random walk

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pages: 369 words: 128,349

Beyond the Random Walk: A Guide to Stock Market Anomalies and Low Risk Investing by Vijay Singal

3Com Palm IPO, Andrei Shleifer, asset allocation, buy and hold, capital asset pricing model, correlation coefficient, cross-subsidies, Daniel Kahneman / Amos Tversky, diversified portfolio, endowment effect, fixed income, index arbitrage, index fund, information asymmetry, liberal capitalism, locking in a profit, Long Term Capital Management, loss aversion, margin call, market friction, market microstructure, mental accounting, merger arbitrage, Myron Scholes, new economy, prediction markets, price stability, profit motive, random walk, Richard Thaler, risk-adjusted returns, risk/return, selection bias, Sharpe ratio, short selling, survivorship bias, transaction costs, Vanguard fund

Using all three quality-identifying proxies simultaneously should improve the detection of price patterns. The idea is that if a stock experiences a large price change unaccompanied by high volume or a public announcement, then that price change may get reversed. On the other hand, a price change accompanied by high volume and a public announcement is likely to be permanent. Indeed, if large price changes are accompanied by an increase in volume and a public announcement by the management or analysts, then a price continuation is likely to result in a one-month abnormal return of about 3.5 percent for positive events and –2.25 percent for negative events. A trading strategy set up to exploit these price patterns can earn an abnormal return of 1.25 percent to 3 percent after transaction costs, that is, 15–36 percent annually. 57 58 Beyond the Random Walk Evidence The evidence on the short-term price drift begins with a large price change as an indicator of a strong signal of new information.

Since an individual stock’s volume, like its return, depends on market volume, it is appropriate to adjust the stock’s trading volume by market volume. The results in panel A of Table 4.2 reveal two return patterns. First, price changes that are accompanied by high volume have price continuations. Price increases with high volume are followed by subsequent increases of 0.20 percent and 0.95 percent over five-day 61 62 Beyond the Random Walk Table 4.2 Returns Following Large Price Changes, High Volume, and News Price Change High Volume Public News Sample Size Abnormal Return: Days 1–5 (%) Abnormal Return: Days 1–20 (%) Panel A: Large price changes with and without high volume 1. 2. 3. 4. Increase Increase Decrease Decrease Yes No Yes No — — — — 1,477 1,442 1,142 812 0.20 –0.71 –0.66 0.49 0.95 –0.67 –0.65 –0.13 Panel B: Large price changes, high volume, and public news 5. 6. 7. 8. Increase Increase Decrease Decrease Yes Yes Yes Yes Yes No Yes No 603 874 653 489 0.54 –0.09 –1.26 0.63 1.98 0.03 –1.68 0.52 Large price change is based on relative return.

Second, the explanation requires a positive return after both positive and negative events in order to compensate for the extra risk. The continuation of the negative price drift after price declines is inconsistent with this explanation. Finally, prices may not react fully to new information because of investor behavior (see Chapter 12 for a discussion). As investors are slow to change their deeply held beliefs about a stock’s value, they tend to underreact. According to this explanation, as investors realize their mistakes, they trade, but with a time lag, resulting in the 67 68 Beyond the Random Walk price drift. In addition, investors are averse to selling at a loss and will continue to delay selling in the hope that the stock price will recover. As a result of this characteristic of investor behavior, the negative drift in price would be less than the positive drift in price. This prediction is supported by the data in Table 4.3.


pages: 209 words: 13,138

Empirical Market Microstructure: The Institutions, Economics and Econometrics of Securities Trading by Joel Hasbrouck

Alvin Roth, barriers to entry, business cycle, conceptual framework, correlation coefficient, discrete time, disintermediation, distributed generation, experimental economics, financial intermediation, index arbitrage, information asymmetry, interest rate swap, inventory management, market clearing, market design, market friction, market microstructure, martingale, price discovery process, price discrimination, quantitative trading / quantitative finance, random walk, Richard Thaler, second-price auction, selection bias, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, two-sided market, ultimatum game, zero-sum game

Later chapters explore multivariate extensions and cointegration. In macroeconomics applications, random-walk decompositions are usually called permanent/transitory. The random-walk terminology is used here to stress the financial economics connection to the random-walk efficient prices. The permanent/transitory distinction is in some respects more descriptive, however, of the attributions that we’re actually making. From a microstructure perspective, the key results expand on those demonstrated for the generalized Roll model: The moving average representation for the price changes suffices to identify the variance of the 2 , the projection of the efficient price on past price implicit efficient price σw changes, and a lower bound on the variance of the difference between the transaction price and the efficient price. The development in this section is heuristic and intuitive.

The discussion then turns to bid and ask quotes, order arrivals, and the resulting transaction price process. 3.2 The Random-Walk Model of Security Prices Before financial economists began to concentrate on the trading process, the standard statistical model for a security price was the random walk. THE ROLL MODEL OF TRADE PRICES The random-walk model is no longer considered to be a complete and valid description of short-term price dynamics, but it nevertheless retains an important role as a model for the fundamental security value. Furthermore, some of the lessons learned from early statistical tests of the randomwalk hypothesis have ongoing relevance in modeling market data. Let pt denote the transaction price at time t, where t indexes regular points of real (“calendar” or “wall-clock”) time, for example, end-of-day, end-of-hour, and so on.

Equation (10.13) suggests a recursive procedure for forecasting. To construct impulse response functions, we set lagged price changes and disturbances to zero and work forward from an initial specified shock εt . By successively computing the impulse response functions for unit shocks in each variable, we may obtain the VMA representation for the price changes pt = θ(L)εt . We may posit for the n prices a random-walk decomposition of the form pt = mt × ι + st (n×1) (n×1) (n×1) where mt = mt−1 + wt , (10.15) where ι is a vector of ones. It is important to note that mt is a scalar: The random-walk (“efficient price”) component is the same for all prices in 2 = [θ(1)] [θ(1)]′ , where [ · ] the model. The random-walk variance is: σw 1 1 1 denotes the first row of the matrix. One property developed earlier for the structural model, however, is general: The rows of θ(1) are identical.


pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Albert Einstein, asset allocation, beat the dealer, Black-Scholes formula, Brownian motion, butterfly effect, buy and hold, capital asset pricing model, collateralized debt obligation, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discrete time, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, iterative process, lateral thinking, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, quantitative trading / quantitative finance, random walk, regulatory arbitrage, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, transaction costs, urban planning, value at risk, volatility arbitrage, volatility smile, Wiener process, yield curve, zero-coupon bond

BM is a very simple yet very rich process, extremely useful for representing many random processes especially those in finance. Its simplicity allows calculations and analysis that would not be possible with other processes. For example, in option pricing it results in simple closed-form formulæ for the prices of vanilla options. It can be used as a building block for random walks with characteristics beyond those of BM itself. For example, it is used in the modelling of interest rates via mean-reverting random walks. Higher-dimensional versions of BM can be used to represent multi-factor random walks, such as stock prices under stochastic volatility. One of the unfortunate features of BM is that it gives returns distributions with tails that are unrealistically shallow. In practice, asset returns have tails that are much fatter than that given by the normal distribution of BM.

• Finding the value of derivatives by exploiting the theoretical relationship between option values and expected payoff under a risk-neutral random walk. Exploring portfolio statistics The most successful quantitative models represent investments as random walks. There is a whole mathematical theory behind these models, but to appreciate the role they play in portfolio analysis you just need to understand three simple concepts. First, you need an algorithm for how the most basic investments evolve randomly. In equities this is often the lognormal random walk. (If you know about the real/risk-neutral distinction then you should know that you will be using the real random walk here.) This can be represented on a spreadsheet or in code as how a stock price changes from one period to the next by adding on a random return. In the fixed-income world you may be using the BGM model to model how interest rates of various maturities evolve.

You would expect equity prices to follow a random walk around an exponentially growing average. So take the logarithm of the stock price and you might expect that to be normal about some mean. That is the non-mathematical explanation for the appearance of the lognormal distribution. More mathematically we could argue for lognormality via the Central Limit Theorem. Using Ri to represent the random return on a stock price from day i - 1 to day i we haveS1 = S0(1 + R1), the stock price grows by the return from day zero, its starting value, to day 1. After the second day we also haveS2 = S1(1 + R2 ) = S0(1 + R1)(1 + R2). After n days we have Figure 2-12: The probability density function for the lognormal random walk evolving through time. the stock price is the initial value multiplied by n factors, the factors being one plus the random returns.


pages: 543 words: 153,550

Model Thinker: What You Need to Know to Make Data Work for You by Scott E. Page

"Robert Solow", Airbnb, Albert Einstein, Alfred Russel Wallace, algorithmic trading, Alvin Roth, assortative mating, Bernie Madoff, bitcoin, Black Swan, blockchain, business cycle, Capital in the Twenty-First Century by Thomas Piketty, Checklist Manifesto, computer age, corporate governance, correlation does not imply causation, cuban missile crisis, deliberate practice, discrete time, distributed ledger, en.wikipedia.org, Estimating the Reproducibility of Psychological Science, Everything should be made as simple as possible, experimental economics, first-price auction, Flash crash, Geoffrey West, Santa Fe Institute, germ theory of disease, Gini coefficient, High speed trading, impulse control, income inequality, Isaac Newton, John von Neumann, Kenneth Rogoff, knowledge economy, knowledge worker, Long Term Capital Management, loss aversion, low skilled workers, Mark Zuckerberg, market design, meta analysis, meta-analysis, money market fund, Nash equilibrium, natural language processing, Network effects, p-value, Pareto efficiency, pattern recognition, Paul Erdős, Paul Samuelson, phenotype, pre–internet, prisoner's dilemma, race to the bottom, random walk, randomized controlled trial, Richard Feynman, Richard Thaler, school choice, sealed-bid auction, second-price auction, selection bias, six sigma, social graph, spectrum auction, statistical model, Stephen Hawking, Supply of New York City Cabdrivers, The Bell Curve by Richard Herrnstein and Charles Murray, The Great Moderation, The Rise and Fall of American Growth, the rule of 72, the scientific method, The Spirit Level, The Wisdom of Crowds, Thomas Malthus, Thorstein Veblen, urban sprawl, value at risk, web application, winner-take-all economy, zero-sum game

Random Walks and Efficient Markets Stock prices prove to be nearly normal random walks with a positive drift to capture gains in the market. Many individual stock prices also are approximately random. Figure 13.3 shows the daily stock price data for Facebook for the year following its initial public offering on May 18, 2012. Facebook was offered at $42 per share. By June 1, 2012, the price had fallen to $28.89. One year later the price had fallen to $24.63. The figure also shows a random walk calibrated to have similar variation. Figure 13.3: Facebook Daily Stock Price June 2012–June 2013 vs. a Random Walk We can apply statistical tests to the sequence of Facebook share prices to see if it satisfies the assumptions of a normal random walk. First, the price should go up and down with equal probability. In the 249 trading days covered, Facebook’s stock price went down on 127 days, or 51% of the time.

Others extended this thinking to create the efficient market hypothesis, which states that at any moment in time the price of a stock captures all relevant information, and future prices must follow a random walk. The efficient market hypothesis rests on paradoxical logic.18 Determining an accurate price requires time and effort. A financial analyst must gather data and construct models. If prices followed a random walk, those activities would have no expected return. However, if no one expends effort to estimate prices, then prices will become inaccurate and the sidewalk will be covered in hundred-dollar bills. In brief, the Grossman and Stiglitz paradox states that if investors believe in the efficient market hypothesis, they stop analyzing, making markets inefficient. If investors believe the market is inefficient, then they perform analyses by applying models, making markets efficient. In point of fact, price movements are rather close to random walks, although sophisticated statistical techniques do reveal short-run patterns.19 While there may be no hundred-dollar bills on the sidewalk, there are some four-leaf clovers in grassy fields that one can find by looking hard enough.

Figure 13.2: Random Walks on Networks Figure 13.2 shows two networks. The network on the left has three nodes forming a triangle. The network on the right has six nodes forming two triangles. We can start a random walk on the left network at A. Suppose that it moves to B, then C, and back to A. The random walk returns to its starting point in three steps. On the network on the right, a random walk starting at D might follow a seven-step path F − G − H − F − E − F − D. If we repeat these experiments many times, the average return times on the network on the left will be shorter. Though unnecessary for small networks such as these, this method becomes useful on larger networks, like the World Wide Web or large email networks. Random Walks and Efficient Markets Stock prices prove to be nearly normal random walks with a positive drift to capture gains in the market.


pages: 461 words: 128,421

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street by Justin Fox

activist fund / activist shareholder / activist investor, Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, bank run, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Bretton Woods, Brownian motion, business cycle, buy and hold, capital asset pricing model, card file, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discovery of the americas, diversification, diversified portfolio, Edward Glaeser, Edward Thorp, endowment effect, Eugene Fama: efficient market hypothesis, experimental economics, financial innovation, Financial Instability Hypothesis, fixed income, floating exchange rates, George Akerlof, Henri Poincaré, Hyman Minsky, implied volatility, impulse control, index arbitrage, index card, index fund, information asymmetry, invisible hand, Isaac Newton, John Meriwether, John Nash: game theory, John von Neumann, joint-stock company, Joseph Schumpeter, Kenneth Arrow, libertarian paternalism, linear programming, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market design, Myron Scholes, New Journalism, Nikolai Kondratiev, Paul Lévy, Paul Samuelson, pension reform, performance metric, Ponzi scheme, prediction markets, pushing on a string, quantitative trading / quantitative finance, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, road to serfdom, Robert Bork, Robert Shiller, Robert Shiller, rolodex, Ronald Reagan, shareholder value, Sharpe ratio, short selling, side project, Silicon Valley, Social Responsibility of Business Is to Increase Its Profits, South Sea Bubble, statistical model, stocks for the long run, The Chicago School, The Myth of the Rational Market, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Kuhn: the structure of scientific revolutions, Thomas L Friedman, Thorstein Veblen, Tobin tax, transaction costs, tulip mania, value at risk, Vanguard fund, Vilfredo Pareto, volatility smile, Yogi Berra

Morgenstern wasn’t enough of a mathematician to do this himself, but he hired young British statistician Clive Granger and put him to work examining stock prices. In 1963, Morgenstern and Granger published a paper confirming that, according to their tests, stock prices moved in a short-term random walk (over the longer run, the movements didn’t look quite so random).13 Morgenstern had connections at Fortune that dated back to the magazine’s coverage of game theory fifteen years before, and his was thus the first of the random walk papers to receive attention in the mainstream press. The headline of the brief item in the magazine’s personal investing section in February 1963 was “A Random Walk in Wall Street.”14 THE HEADQUARTERS OF THIS EARLY random walk movement was Samuelson’s MIT. The university’s new Sloan School of Industrial Management shared a building with the Economics Department.

Over the forty-two years of data he examined, Working found that the speculators had, as a group, lost money.32 Moving on, Working began to study the movements of futures prices. He found a few interesting patterns. “Wheat prices tend strongly to rise during a season following three of low average price and to decline during a season following three of high average price,” he reported in 1931. “The relation is attributed partly to a tendency for price judgments of wheat traders to be unduly influenced by memory of prices in recent years.”33 Much of what Working saw in price movements, though, seemed random. The phrase “random walk” appears to have been coined in 1905, in an exchange in the letters pages of the English journal Nature concerning the mathematical description of the meanderings of a hypothetical drunkard.34 Most early studies of economic data had been a search not for drunken meanderings but for recognizable patterns and, not surprisingly, many were found.

“It’s easier to win than lose,” Business Week, May 29, 1965, 122. 20. Arnold Moore, interview with the author. Moore finished his dissertation in 1962, and it was published as “Some Characteristics of Changes in Common Stock Prices,” in The Random Character of Stock Prices, Paul Cootner, ed. (Cambridge, Mass.: MIT Press, 1964), 139–61. 21. Robert A. Levy, “Random Walks: Reality or Myth,” Financial Analysts Journal (Nov.–Dec. 1967): 69–77. 22. Michael C. Jensen, “Random Walks: Reality or Myth—Comment,” Financial Analysts Journal (Nov.–Dec. 1967): 84. 23. Jensen, “Random Walks,” 81. 24. Eugene F. Fama, Lawrence Fisher, Michael C. Jensen, Richard Roll, “The Adjustment of Stock Prices to New Information,” International Economic Review (Feb. 1969): 1–21. It took the paper years to get to print because Fama was set on publishing it somewhere other than the Chicago Business School’s Journal of Business, where all his previous papers had ended up, and it was a struggle to find another journal willing to take it.


pages: 482 words: 121,672

A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing (Eleventh Edition) by Burton G. Malkiel

accounting loophole / creative accounting, Albert Einstein, asset allocation, asset-backed security, beat the dealer, Bernie Madoff, bitcoin, butter production in bangladesh, buttonwood tree, buy and hold, capital asset pricing model, compound rate of return, correlation coefficient, Credit Default Swap, Daniel Kahneman / Amos Tversky, Detroit bankruptcy, diversification, diversified portfolio, dogs of the Dow, Edward Thorp, Elliott wave, Eugene Fama: efficient market hypothesis, experimental subject, feminist movement, financial innovation, financial repression, fixed income, framing effect, George Santayana, hindsight bias, Home mortgage interest deduction, index fund, invisible hand, Isaac Newton, Long Term Capital Management, loss aversion, margin call, market bubble, money market fund, mortgage tax deduction, new economy, Own Your Own Home, passive investing, Paul Samuelson, pets.com, Ponzi scheme, price stability, profit maximization, publish or perish, purchasing power parity, RAND corporation, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, short selling, Silicon Valley, South Sea Bubble, stocks for the long run, survivorship bias, the rule of 72, The Wisdom of Crowds, transaction costs, Vanguard fund, zero-coupon bond, zero-sum game

Stocks are likened to fullbacks who, once having gained some momentum, can be expected to carry on for a long gain. It turns out that this is simply not the case. Sometimes one gets positive price changes (rising prices) for several days in a row; but sometimes when you are flipping a fair coin you also get a long string of “heads” in a row, and you get sequences of positive (or negative) price changes no more frequently than you can expect random sequences of heads or tails in a row. What are often called “persistent patterns” in the stock market occur no more frequently than the runs of luck in the fortunes of any gambler. This is what economists mean when they say that stock prices behave very much like a random walk. JUST WHAT EXACTLY IS A RANDOM WALK? To many people this appears to be errant nonsense. Even the most casual reader of the financial pages can easily spot patterns in the market.

Slow adjustment can make stock prices rise steadily for a period, imparting a degree of momentum. The failure of stock prices to measure up perfectly to the definition of a random walk led the financial economists Andrew Lo and A. Craig MacKinlay to publish a book entitled A Non-Random Walk Down Wall Street. In addition to some evidence of short-term momentum, there has been a long-run uptrend in most averages of stock prices in line with the long-run growth of earnings and dividends. But don’t count on short-term momentum to give you some surefire strategy to allow you to beat the market. For one thing, stock prices don’t always underreact to news—sometimes they overreact and price reversals can occur with terrifying suddenness. We shall see in chapter 11 that investment funds managed in accordance with a momentum strategy started off with subpar results.

Participants were faced with a choice—sell their cards at some negotiated price or hold on to them and hope to win. Obviously, each card had the same probability of winning. Nevertheless, the prices at which players were willing to sell their cards were systematically higher for those who chose their cards than for the group who had simply been given a card. Insights such as this led to the decision to let state lottery buyers pick their own numbers even though luck alone determines lottery winners. It is this illusion of control that can lead investors to see trends that do not exist or to believe that they can spot a stock-price pattern that will predict future prices. In fact, despite considerable efforts to tease some form of predictability out of stock-price data, the development of stock prices from period to period is very close to a random walk, where price changes in the future are essentially unrelated to changes in the past.


pages: 338 words: 106,936

The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business cycle, butterfly effect, buy and hold, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, Vilfredo Pareto, volatility smile

To see how this kind of mathematics can be helpful in understanding financial markets, you just have to see that a stock price is a lot like our man in Cancun. At any instant, there is a chance that the price will go up, and a chance that the price will go down. These two possibilities are directly analogous to the drunkard stumbling toward room 700, or toward room 799, working his way up or down the hallway. And so, the question that mathematics can answer in this case is the following: If the stock begins at a certain price, and it undergoes a random walk, what is the probability that the price will be a particular value after some fixed period of time? In other words, which door will the price have stumbled to after one hundred, or one thousand, ticks? This is the question Bachelier answered in his thesis. He showed that if a stock price undergoes a random walk, the probability of its taking any given value after a certain period of time is given by a curve known as a normal distribution, or a bell curve.

As its name suggests, this curve looks like a bell, rounded at the top and widening at the bottom. The tallest part of this curve is centered at the starting price, which means that the most likely scenario is that the price will be somewhere near where it began. Farther out from this center peak, the curve drops off quickly, indicating that large changes in price are less likely. As the stock price takes more steps on the random walk, however, the curve progressively widens and becomes less tall overall, indicating that over time, the chances that the stock will vary from its initial value increase. A picture is priceless here, so look at Figure 1 to see how this works. Figure 1: Bachelier discovered that if the price of a stock undergoes a random walk, the probability that the price will take a particular value in the future can be calculated from a curve known as a normal distribution.

By the time the Cootner book was published in 1964, the idea that market prices follow a random walk was well entrenched, and many economists recognized that Bachelier was responsible for it. But the random walk model wasn’t the punch line of Bachelier’s thesis. He thought of it as preliminary work in the service of his real goal, which was developing a model for pricing options. An option is a kind of derivative that gives the person who owns the option the right to buy (or sometimes sell) a specific security, such as a stock or bond, at a predetermined price (called the strike price), at some future time (the expiration date). When you buy an option, you don’t buy the underlying stock directly. You buy the right to trade that stock at some point in the future, but at a price that you agree to in the present. So the price of an option should correspond to the value of the right to buy something at some time in the future.


pages: 416 words: 118,592

A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing by Burton G. Malkiel

accounting loophole / creative accounting, Albert Einstein, asset allocation, asset-backed security, backtesting, beat the dealer, Bernie Madoff, BRICs, butter production in bangladesh, buy and hold, capital asset pricing model, compound rate of return, correlation coefficient, Credit Default Swap, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, dogs of the Dow, Edward Thorp, Elliott wave, Eugene Fama: efficient market hypothesis, experimental subject, feminist movement, financial innovation, fixed income, framing effect, hindsight bias, Home mortgage interest deduction, index fund, invisible hand, Isaac Newton, Long Term Capital Management, loss aversion, margin call, market bubble, money market fund, mortgage tax deduction, new economy, Own Your Own Home, passive investing, Paul Samuelson, pets.com, Ponzi scheme, price stability, profit maximization, publish or perish, purchasing power parity, RAND corporation, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, short selling, Silicon Valley, South Sea Bubble, stocks for the long run, survivorship bias, The Myth of the Rational Market, the rule of 72, The Wisdom of Crowds, transaction costs, Vanguard fund, zero-coupon bond

Slow adjustment can make stock prices rise steadily for a period, imparting a degree of momentum. The failure of stock prices to measure up perfectly to the definition of a random walk led the financial economists Andrew Lo and A. Craig MacKinlay to publish a book entitled A Non-Random Walk Down Wall Street. In addition to some evidence of short-term momentum, there has been a long-run uptrend in most averages of stock prices in line with the long-run growth of earnings and dividends. But don’t count on short-term momentum to give you some sure-fire strategy to allow you to beat the market. For one thing, stock prices don’t always underreact to news—sometimes they overreact and price reversals can occur with terrifying suddenness. Two mutual funds managed in accordance with a momentum strategy started off with distinctly subpar returns. And even during periods when momentum is present (and the market fails to behave like a random walk), the systematic relationships that exist are often so small that they are not useful to investors.

Participants were faced with a choice—sell their cards at some negotiated price or hold on to them and hope to win. Obviously, each card had the same probability of winning. Nevertheless, the prices at which players were willing to sell their cards were systematically higher for those who chose their cards than for the group who had simply been given a card. Insights such as this led to the decision to let state lottery buyers pick their own numbers even though luck alone determines lottery winners. It is this illusion of control that can lead investors to see trends that do not exist or to believe that they can spot a stock-price pattern that will predict future prices. In fact, despite considerable efforts to tease some form of predictability out of stock-price data, the development of stock prices from period to period is very close to a random walk, where price changes in the future are essentially unrelated to changes in the past.

The life-cycle investment guide described in Part Four gives individuals of all age groups specific portfolio recommendations for meeting their financial goals, including advice on how to invest in retirement. WHAT IS A RANDOM WALK? A random walk is one in which future steps or directions cannot be predicted on the basis of past history. When the term is applied to the stock market, it means that short-run changes in stock prices are unpredictable. Investment advisory services, earnings forecasts, and complicated chart patterns are useless. On Wall Street, the term “random walk” is an obscenity. It is an epithet coined by the academic world and hurled insultingly at the professional soothsayers. Taken to its logical extreme, it means that a blindfolded monkey throwing darts at the stock listings could select a portfolio that would do just as well as one selected by the experts.


pages: 733 words: 179,391

Adaptive Markets: Financial Evolution at the Speed of Thought by Andrew W. Lo

"Robert Solow", Albert Einstein, Alfred Russel Wallace, algorithmic trading, Andrei Shleifer, Arthur Eddington, Asian financial crisis, asset allocation, asset-backed security, backtesting, bank run, barriers to entry, Berlin Wall, Bernie Madoff, bitcoin, Bonfire of the Vanities, bonus culture, break the buck, Brownian motion, business cycle, business process, butterfly effect, buy and hold, capital asset pricing model, Captain Sullenberger Hudson, Carmen Reinhart, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, corporate governance, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, cryptocurrency, Daniel Kahneman / Amos Tversky, delayed gratification, Diane Coyle, diversification, diversified portfolio, double helix, easy for humans, difficult for computers, Ernest Rutherford, Eugene Fama: efficient market hypothesis, experimental economics, experimental subject, Fall of the Berlin Wall, financial deregulation, financial innovation, financial intermediation, fixed income, Flash crash, Fractional reserve banking, framing effect, Gordon Gekko, greed is good, Hans Rosling, Henri Poincaré, high net worth, housing crisis, incomplete markets, index fund, interest rate derivative, invention of the telegraph, Isaac Newton, James Watt: steam engine, job satisfaction, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Meriwether, Joseph Schumpeter, Kenneth Rogoff, London Interbank Offered Rate, Long Term Capital Management, longitudinal study, loss aversion, Louis Pasteur, mandelbrot fractal, margin call, Mark Zuckerberg, market fundamentalism, martingale, merger arbitrage, meta analysis, meta-analysis, Milgram experiment, money market fund, moral hazard, Myron Scholes, Nick Leeson, old-boy network, out of africa, p-value, paper trading, passive investing, Paul Lévy, Paul Samuelson, Ponzi scheme, predatory finance, prediction markets, price discovery process, profit maximization, profit motive, quantitative hedge fund, quantitative trading / quantitative finance, RAND corporation, random walk, randomized controlled trial, Renaissance Technologies, Richard Feynman, Richard Feynman: Challenger O-ring, risk tolerance, Robert Shiller, Robert Shiller, Sam Peltzman, Shai Danziger, short selling, sovereign wealth fund, Stanford marshmallow experiment, Stanford prison experiment, statistical arbitrage, Steven Pinker, stochastic process, stocks for the long run, survivorship bias, Thales and the olive presses, The Great Moderation, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, theory of mind, Thomas Malthus, Thorstein Veblen, Tobin tax, too big to fail, transaction costs, Triangle Shirtwaist Factory, ultimatum game, Upton Sinclair, US Airways Flight 1549, Walter Mischel, Watson beat the top human players on Jeopardy!, WikiLeaks, Yogi Berra, zero-sum game

• 21 Samuelson spotted a paradox: if the weather influenced the price of grain, how could the price of grain follow a random walk?20 Samuelson knew that weather patterns, while complicated, did not behave randomly, and certainly the seasons didn’t follow each other randomly either. It seemed to Samuelson that Bachelier’s Random Walk actually proved too much. Samuelson resolved this difficulty in a very quick and elegant way, characteristic of his personal style in economics. Using the mathematical technique of induction, Samuelson showed that all the information of an asset’s past price changes are bundled in the asset’s present price. The price already contains all the known information about the asset up to that point—changes in the weather, storage costs, etc. Everything has already been taken into account. As a result, past price changes carry no information in predicting the asset’s next price.

As a result, past price changes carry no information in predicting the asset’s next price. Samuelson reasoned as follows. If investors were able to incorporate all the potential impact of future events on an asset’s price today, then future price changes could not be predicted based on any of today’s information. If they could, investors would have used that information in the first place. As a result, prices must move unpredictably. If a market is informationally efficient—that is, if prices fully incorporate the expectations of all the players in the market—then the following price changes will necessarily be impossible to forecast. It’s a subtle idea, but it’s clearly related to Cardano’s martingale and Bachelier’s random walk. The title of Samuelson’s seminal 1965 article neatly summarizes his main idea: “Proof that Properly Anticipated Prices Fluctuate Randomly,” but we know it better today as the Efficient Markets Hypothesis.21 The Efficient Markets Hypothesis seemed so simpleminded to Samuelson that he withheld publishing it for years.

Despite our best efforts, however, we were unable to explain away the evidence against the Random Walk Hypothesis. At first, we thought our results might be due to the fact that we used weekly returns, since prior studies that supported the Random Walk Hypothesis used daily returns. But we soon discovered that the case against the random walk was equally persuasive with daily returns. We looked into possible sources of bias in the market data itself, such as subtle errors introduced by incorrectly assuming that all closing prices occur at the same time of day. (An active stock like Apple will trade until the closing bell, 4:00 p.m. Eastern Standard Time, while the last trade of, say, Koffee Meister might occur at 3:55 p.m.) We investigated the effect of price discreteness: in those days, stock prices moved in “ticks” of an eighth of a dollar ($0.125), which can create some interesting but spurious patterns in prices.


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Why Stock Markets Crash: Critical Events in Complex Financial Systems by Didier Sornette

Asian financial crisis, asset allocation, Berlin Wall, Bretton Woods, Brownian motion, business cycle, buy and hold, capital asset pricing model, capital controls, continuous double auction, currency peg, Deng Xiaoping, discrete time, diversified portfolio, Elliott wave, Erdős number, experimental economics, financial innovation, floating exchange rates, frictionless, frictionless market, full employment, global village, implied volatility, index fund, information asymmetry, intangible asset, invisible hand, John von Neumann, joint-stock company, law of one price, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, market design, market fundamentalism, mental accounting, moral hazard, Network effects, new economy, oil shock, open economy, pattern recognition, Paul Erdős, Paul Samuelson, quantitative trading / quantitative finance, random walk, risk/return, Ronald Reagan, Schrödinger's Cat, selection bias, short selling, Silicon Valley, South Sea Bubble, statistical model, stochastic process, stocks for the long run, Tacoma Narrows Bridge, technological singularity, The Coming Technological Singularity, The Wealth of Nations by Adam Smith, Tobin tax, total factor productivity, transaction costs, tulip mania, VA Linux, Y2K, yield curve

This corresponds to a specific realization of the random numbers used in generating the random walks W t represented in the second panel. The top panel is obtained by taking a power of the inverse of a constant Wc , here taken equal to 1 minus the random walk shown in the second panel. In this case, when the random walk approaches 1, the bubble diverges. Notice the similarity between the trajectories shown in the top (Bt) and second (W t) panels as long as the random walk W t does not approach the value Wc = 1 too much. It is free to wander, but when it approaches 1, the bubble price Bt shows much greater sensitivity and eventually diverges when W t reaches 1. Before this happens, Bt can exhibit local peaks, that is, local bubbles, which come back smoothly. This corresponds to realizations of when the random walk approaches Wc without touching it and then spontaneously recedes away from it.

We can conclude that the residual correlations are those little enough not to be profitable by strategies such as those described above due to “imperfect” market conditions. In other words, the liquidity and efficiency of markets control the degree of correlation that is compatible with a near absence of arbitrage opportunity. THE EFFICIENT MARKET HYPOTHESIS AND THE RANDOM WALK Such observations have been made for a long time. A pillar of modern finance is the 1900 Ph.D. thesis dissertation of Louis Bachelier, in Paris, and his subsequent work, especially in 1906 and 1913 [25]. To account for the apparent erratic motion of stock market prices, he proposed that price trajectories are identical to random walks. The Random Walk The concept of a random walk is simple but rich for its many applications, not only in finance but also in physics and the description of natural phenomena. It is arguably one of the most important founding concepts in modern physics as well as in finance, as it underlies the theories of elementary particles, which are the building blocks of our universe, as well as those describing the complex organization of matter around us.

Figure 5.8 shows a typical trajectory of the bubble component of the price generated by the nonlinear positive feedback model [396], starting from some initial value up to the time just before the price starts to blow up. The simplest version of this model consists in a bubble price Bt being essentially a power of the inverse of a random walk W t in the following sense. Starting from B0 = W 0 = 0 at the origin of time, when the random walk approaches some value Wc , here 165 model ing bubbles a n d c r a s h e s B(t) 4.0 2.0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 W(t) 1.0 0.5 0 dB(t) 0.2 0 -0.2 dW(t) 0.1 0 -0.1 t Fig. 5.8. Top panel: Realization of a bubble price Bt as a function of time constructed from the “singular inverse random walk.” This corresponds to a specific realization of the random numbers used in generating the random walks W t represented in the second panel.


The Armchair Economist: Economics and Everyday Life by Steven E. Landsburg

Albert Einstein, Arthur Eddington, business cycle, diversified portfolio, first-price auction, German hyperinflation, Golden Gate Park, information asymmetry, invisible hand, Kenneth Arrow, means of production, price discrimination, profit maximization, Ralph Nader, random walk, Ronald Coase, Sam Peltzman, sealed-bid auction, second-price auction, second-price sealed-bid, statistical model, the scientific method, Unsafe at Any Speed

I had misinterpreted the word random to mean "unrelated to anything else in the world," which is why I thought that the random walk theory denied that IBM's behavior could affect its stock price. But one random event can be perfectly correlated with another. Great corporate blunders arrive randomly, and the corresponding stock price changes arrive along with them. Economists believe that stock market prices behave a lot like random walks most of the time. That is, we believe that price changes (not prices) usually have the same statistical characteristics as the series of numbers generated by a roulette wheel. If prices were random, as I once erroneously believed, then today's price would be useless as a predictor of tomorrow's. Because price changes are random, the opposite is true. Today's price is the best possible predictor of tomorrow's. Tomorrow's price is today's price, plus a (usually small) random adjustment. 190 HOW MARKETS WORK Imagine a simple game of chance.

I'll be in my well-worn armchair, thinking about things. CHAPTER 20 RANDOM WALKS AND STOCK MARKET PRICES A Primer for Investors When I was young and first heard that stock market prices follow random walks, I was incredulous. Did this mean that IBM might as well replace its corporate officers with underprivileged eight-year-olds? My question was born of naivete', and of considerable ignorance. I've learned a lot in the interim. One thing I've learned is that a random walk is not a theory of prices; it is a theory of price changes. In that distinction lies a world of difference. My original (entirely wrong) conception invoked a roulette wheel as its central image. One day the little ball lands on 10, and the stock price is $10; the next day it lands on 8, and the price falls to $8, or it lands on 20 and the stockholders get rich.

Or that because it has recently risen, it is likely to fall soon or to rise further. But if stock prices are like random walks, as economists believe they usually are, then future price changes are quite independent of past history. The current price predicts the future price. The commentators notwithstanding, past price changes predict nothing. Those who play the market like to believe that they are more sophisticated than those who play casino games. Yet only the most naive roulette player would suggest that because his cash balance has fallen over the last several plays, it is now due for a "correction" upward. Experienced gamblers know what to expect from a random walk. When I was young, I harbored many misconceptions (not all of them related to finance). Another was that in the presence of a random walk, there can be no role for investment strategy.


Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals by David Aronson

Albert Einstein, Andrew Wiles, asset allocation, availability heuristic, backtesting, Black Swan, butter production in bangladesh, buy and hold, capital asset pricing model, cognitive dissonance, compound rate of return, computerized trading, Daniel Kahneman / Amos Tversky, distributed generation, Elliott wave, en.wikipedia.org, feminist movement, hindsight bias, index fund, invention of the telescope, invisible hand, Long Term Capital Management, mental accounting, meta analysis, meta-analysis, p-value, pattern recognition, Paul Samuelson, Ponzi scheme, price anchoring, price stability, quantitative trading / quantitative finance, Ralph Nelson Elliott, random walk, retrograde motion, revision control, risk tolerance, risk-adjusted returns, riskless arbitrage, Robert Shiller, Robert Shiller, Sharpe ratio, short selling, source of truth, statistical model, stocks for the long run, systematic trading, the scientific method, transfer pricing, unbiased observer, yield curve, Yogi Berra

First, “the usual method of graphing stock prices gives a picture of successive (price) levels rather than of price changes and levels can give an artificial appearance of pattern or trend. Second, chance behavior itself produces patterns that invite spurious interpretations.”128 Roberts showed that the same chart patterns to which TA attaches importance129 appear with great regularity in random walks. A random walk is, by definition, devoid of authentic trends, patterns, or exploitable order of any kind. However, Roberts’ random-walk charts displayed headand-shoulder tops and bottoms, triangle tops and bottoms, triple tops and bottoms, trend channels, and so forth. You can create a random-walk chart from a sequence of coin flips by starting with an arbitrary price, say $100, and adding one dollar for each head and subtracting one dollar for each tail.

It occurs when an observer has no prior belief about whether the process generating the data is random or nonrandom The clustering illusion is the misperception of order (nonrandomness) in data that is actually a random walk. Again, imagine someone observing the outcomes of a process that is truly a random walk trying to determine if the process is random or nonrandom (orderly, systematic).71 Recall that small samples of random walks often appear more trended (clustered) than common sense would lead us to expect (the hot hand in basketball). As a result of the clustering illusion, a sequence of positive price or earnings changes is wrongly interpreted as a legitimate trend, when it is nothing more than an ordinary streak in a random walk. Social Factors: Imitative Behavior, Herding, and Information Cascades72 We have just seen how investor behavior viewed at the level of the individual investor can explain several types of systematic price movement. This section examines investor behavior at the group level to explain systematic price movements.

So it is hard to imagine that the market as a whole reflects the emotions described by these psychological theories.”7 We will need to look beyond the platitudes of popular texts for TA’s justification. Fortunately, theories developed in the field of behavioral finance and elsewhere are beginning to offer the theoretical support TA needs. THE ENEMY’S POSITION: EFFICIENT MARKETS AND RANDOM WALKS Before discussing theories that explain why nonrandom price movements should exist, we need to consider the enemy’s position, the EMH. Recently, some have argued that EMH does not necessarily imply that prices follow unpredictable random walks,8 and that efficient markets and price predictability can coexist. However, the pioneers of EMH asserted that random walks were a necessary consequence of efficient markets. This section states their case and examines its weaknesses. What Is an Efficient Market? An efficient market is a market that cannot be beaten. In such a market, no fundamental or technical analysis strategy, formula, or system can earn a risk-adjusted rate of return that beats the market defined by a benchmark index.


Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

Asian financial crisis, asset allocation, backtesting, buy and hold, capital asset pricing model, collateralized debt obligation, commodity trading advisor, compound rate of return, constrained optimization, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, discrete time, distributed generation, diversification, diversified portfolio, dividend-yielding stocks, fixed income, high net worth, implied volatility, index arbitrage, index fund, interest rate swap, iterative process, linear programming, London Interbank Offered Rate, Long Term Capital Management, market fundamentalism, merger arbitrage, Mexican peso crisis / tequila crisis, p-value, Pareto efficiency, Ponzi scheme, quantitative trading / quantitative finance, random walk, risk-adjusted returns, risk/return, selection bias, Sharpe ratio, short selling, stochastic process, survivorship bias, systematic trading, technology bubble, transaction costs, value at risk, zero-sum game

Some authors have suggested that unit root tests suffer from low power and that the test does not discriminate very 328 PROGRAM EVALUATION, SELECTION, AND RETURNS well between mean reverting series and series that do not mean revert at all (Kennedy 1998). However, the robustness of the ADF test is increased when lags are used. If a series is found to be nonstationary by the ADF test, it does not necessarily imply that it behaves like a random walk, because random walks are but one example of nonstationary time series. Fortunately, the ADF test also can be used to test specifically for random walks. No CTA strategy that relies solely on historical prices can be continuously profitable if markets are efficient and the random walk hypothesis holds true. In this case, future percent changes in NAVs would be entirely unrelated by the historical performance (Pindyck and Rubinfeld 1998). Recent studies have shown that a minimal amount of performance persistence is found in CTAs and there could exist some advantages in selecting CTAs based on past performance when a long time series of data is available and accurate methods are used (Brorsen and Townsend 2002).

T INTRODUCTION This chapter investigates whether monthly percent changes in net asset values (NAVs) of commodity trading advisor (CTA) classifications follow random walks. Previous econometric studies of financial time series have employed unit root tests, such as the Augmented Dickey-Fuller test (ADF), to identify random walk behavior in stock prices and market indices, for example. The characteristics of CTAs are such that investment into this alternative investment class can enhance portfolio returns, but these characteristics are likely to be mitigated if pure random walk behavior is present because that would imply a lack of evidence of value added to the portfolio (differential manager skill). Research into the performance persistence of CTAs is sparse, so there is little information on the long-term diligence of these managers (Edwards This article previously appeared in Journal of Alternative Investments, No. 2, 2003.

CTAs assume both long and short market positions, and realize profits when there are persistent trends in markets and when those trends can be identified early enough. Thus, the performance of CTAs depends not only on price movements, but also on the managers’ ability to identify them. One possible explanation for random walk behavior during the examination period is due to the fact that traditional CTAs make large profits during extreme market movements, themselves random events. Their correlations may be more accurate and stable if they are used as a hedge against short volatility exposure. The discretionary, currency, and European traders trade in periods of high liquidity, which has been the case since 1995. We found that only one class, diversified, did not behave as a random walk, likely since trends in a diversified portfolio are stable, although they may not produce sufficient profits to satisfy the expectations of all investors.


pages: 364 words: 101,286

The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot, Richard L. Hudson

Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John Meriwether, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, Myron Scholes, new economy, paper trading, passive investing, Paul Lévy, Paul Samuelson, plutocrats, Plutocrats, price mechanism, quantitative trading / quantitative finance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, Vilfredo Pareto, volatility smile

On average—just as in the coin-toss game—he gets nowhere. So if you consider only that average, his random walk across the field will be forever stuck at his starting point. And that would be the best possible forecast of his future position at any time, if you had to make such a guess. The same reasoning applies to a bond price: In the absence of new information that might change the balance of supply and demand, what is the best possible forecast of the price tomorrow? Again, the price can go up or down, by big increments or small. But, with no new information to push the price decisively in one direction or another, the price on average will fluctuate around its starting point. So again, the best forecast is the price today. Moreover, each variation in price is unrelated to the last, and is generated by the same unchanging but mysterious process that drives the markets.

It takes no great leap of the imagination to see how such spurious patterns could also appear in otherwise random financial data. This is not to say that price charts are meaningless, or that prices all vary by the whim of luck. But it does say that, when examining price charts, we should guard against jumping to conclusions that the invisible hand of Adam Smith is somehow guiding them. It is a bold investor who would try to forecast a specific price level based solely on a pattern in the charts. 9. Forecasting Prices May Be Perilous, but You Can Estimate the Odds of Future Volatility. All is not hopeless. Markets are turbulent, deceptive, prone to bubbles, infested by false trends. It may well be that you cannot forecast prices. But evaluating risk is another matter entirely. Step back a moment. The classic Random Walk model makes three essential claims. First is the so-called martingale condition: that your best guess of tomorrow’s price is today’s price.

What a company does today—a merger, a spin-off, a critical product launch—shapes what the company will look like a decade hence; in the same way, its stock-price movements today will influence movements tomorrow. Others suggest that the market may take a long time to absorb and fully price information. When confronted by bad news, some quick-triggered investors react immediately while others, with different financial goals and longer time-horizons, may not react for another month or year. Whatever the explanation, we can confirm the phenomenon exists—and it contradicts the random-walk model. Second, contrary to orthodoxy, price changes are very far from following the bell curve. If they did, you should be able to run any market’s price records through a computer, analyze the changes, and watch them fall into the approximate “normality” assumed by Bachelier’s random walk. They should cluster about the mean, or average, of no change. In fact, the bell curve fits reality very poorly.


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Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

Albert Einstein, anti-communist, asset allocation, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Brownian motion, buy and hold, buy low sell high, capital asset pricing model, Claude Shannon: information theory, computer age, correlation coefficient, diversified portfolio, Edward Thorp, en.wikipedia.org, Eugene Fama: efficient market hypothesis, high net worth, index fund, interest rate swap, Isaac Newton, Johann Wolfgang von Goethe, John Meriwether, John von Neumann, Kenneth Arrow, Long Term Capital Management, Louis Bachelier, margin call, market bubble, market fundamentalism, Marshall McLuhan, Myron Scholes, New Journalism, Norbert Wiener, offshore financial centre, Paul Samuelson, publish or perish, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, Rubik’s Cube, short selling, speech recognition, statistical arbitrage, The Predators' Ball, The Wealth of Nations by Adam Smith, transaction costs, traveling salesman, value at risk, zero-coupon bond, zero-sum game

The butterfly whose flapping causes a hurricane could lead to the sinking of a yacht full of Sperry executives, pummeling the stock’s price. How can anyone predict such contingencies systematically? Then Thorp thought of the random walk model. Assume that there is no possible way of predicting the events that move stock prices. Then buying a stock option is placing a bet on a random walk. Thorp knew that there were already precise methods for calculating the probability distributions of random walks. They depend on the average size of the random motions—in this case, how much a stock’s price changes, up or down, per day. Thorp did some computations. He found that most warrants were priced like carnival games. They cost too much, given what you can win and your chance of winning it. This was especially true of warrants that were within a couple of years of expiring.

The mathematical treatment of Brownian motion that Einstein published in 1905 was similar to, but less advanced than, the one that Bachelier had already derived for stock prices. Einstein, like practically everyone else, had never heard of Bachelier. The Random Walk Cosa Nostra SAMUELSON ADOPTED Bachelier’s ideas into his own thinking. Characteristically, he did everything he could to acquaint people with Bachelier’s genius. Just as characteristically, Samuelson called Bachelier’s views “ridiculous.” Huh? Samuelson spotted a mistake in Bachelier’s work. Bachelier’s model had failed to consider that stock prices cannot fall below zero. Were stock price changes described by a conventional random walk, it would be possible for prices to wander below zero, ending up negative. That can’t happen in the real world. Investors are protected by limited liability.

No matter what goes wrong with a company, the investors do not end up owing money. This spoiled Bachelier’s neat model. Samuelson found a simple fix. He suggested that each day, a stock’s price is multiplied by a random factor (like 98 or 105 percent) rather than increased or decreased by a random amount. A stock might, for instance, be just as likely to double in price as to halve in price over a certain time frame. This model, called a log-normal or geometric random walk, prevents stocks from taking on negative values. To Samuelson, the random walk suggested that the stock market was a glorified casino. If the daily movements of stock prices are as unpredictable as the daily lotto numbers, then maybe people who make fortunes in the market are like people who win lotteries. They are lucky, not smart. It follows that all the people who advise clients on which stocks to buy are quacks.


The Intelligent Asset Allocator: How to Build Your Portfolio to Maximize Returns and Minimize Risk by William J. Bernstein

asset allocation, backtesting, buy and hold, capital asset pricing model, commoditize, computer age, correlation coefficient, diversification, diversified portfolio, Eugene Fama: efficient market hypothesis, fixed income, index arbitrage, index fund, intangible asset, Long Term Capital Management, p-value, passive investing, prediction markets, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, South Sea Bubble, stocks for the long run, survivorship bias, the rule of 72, the scientific method, time value of money, transaction costs, Vanguard fund, Yogi Berra, zero-coupon bond

.): correlation coefficients, 71–74 international diversification with small stocks, 74–75 risk tolerance and, 79–80, 143 three-step approach to, 75–83 Out of sample, 87 Overbalancing, 138 Overconfidence, 139–140 P/B ratio (See Book value) P/E ratio: data on ranges of, 113, 114 earnings yield as reverse of, 119 in new era of investing, 124 in value investing, 112, 119–120 Pacific Rim stocks, 19, 20, 21, 25, 55–59, 147, 156 Pension funds, 103 (See also Institutional investors) Perfectly reasonable price (PRP), 127–128 Performance measurement: alpha in, 89–90, 98 three-factor model in, 123–124 (See also Benchmarking) Perold, Andre, 141 Persistence of performance, 85–88 Peters, Tom, 118 Piscataqua Research, 103 Policy allocation, 59 Portfolio insurance, 141 Portfolio Selection (Markowitz), 177–178 Precious metals stocks, 19–20, 21, 48, 55, 57, 59 Price, Michael, 162 Professional investors (See Institutional investors) Prudent man test, 60 Random Walk Down Wall Street, A (Malkiel), 101–102, 175 Random walk theory, 106–108, 119 positive autocorrelation and, 106–108 204 Index Random walk theory (Cont.): random walk defined, 106 rebalancing and, 109 Raskob, John J., 16–17 Real estate investment trusts (REITs), 38, 40, 100, 145 defined, 19 index fund, 148 returns on, 19, 21, 25 Real return, 26, 80, 168, 170 Rebalancing: frequency of, 108–109 importance of, 32–33, 35–36, 59, 63, 174 and mean-variance optimizer (MVO), 65 overbalancing in, 138 random walk theory and, 109 rebalancing bonus, 74, 159–160 of tax-sheltered accounts, 159–160 of taxable accounts, 160–161 Recency effects, 47–48, 52, 53, 58–59, 140–141 Regression analysis, 89–90 Reinvestment risk, 23 Representativeness, 118 Research expenses, 92, 95 Residual return, 98 Retirement, 165–172 asset allocation for, 153–154 duration risk and, 165–167 shortfall risk and, 167–172 (See also Tax-sheltered accounts) Return: annualized, 2–3, 5 average, 2–3 coin toss and, 1–5 company size and, 116–117 correlation between risk and, 21 dividend discount method, 23–24, 26, 127–132 efficient frontier and, 55–58 expected investment, 26 historical, problems with, 21–27 Return (Cont.): impact of diversification on, 31–36, 63 market, 168 real, 26, 80, 168, 170 return and risk plot, 31–36, 41–45 risk and high, 18 uncorrelated, 29–31 variation in, 116–117 Risk: common stock, 1–5 correlation between return and, 21 currency, 132–137 duration, 165–167 efficient frontier and, 55–58 excess, 12–13 high returns and, 18 impact of diversification on, 31–36, 63 nonsystematic, 12–13 reinvestment, 23 return and risk plot, 31–36, 41–45 shortfall, 167–172 sovereign, 72 systematic, 13 (See also Standard deviation) Risk aversion myopia, 141–142 Risk dilution, 45–46 Risk-free investments, 10, 15, 152 Risk-free rate, 121 Risk time horizon, 130, 131, 143–144, 167 Risk tolerance, 79–80, 143 Roth IRA, 172 Rukeyser, Lou, 174 Rule of 72, 27 Sanborn, Robert, 88–90 Securities Act of 1933, 92–93 Security Analysis (Graham and Dodd), 93, 118, 125, 176 Selling forward, 132–133 Semivariance, 7 Sharpe, William, 141 Shortfall risk, 167–172 Siegel, Jeremy, 19, 136 Index Simple portfolios, 31–36 Sinquefield, Rex, 148 Small-cap premium, 53, 121, 122 Small-company stocks, 13–16, 25 correlation with large-company stocks, 53–55 efficient frontier and, 55–59 indexing, 101, 102, 148–149 international diversification with, 74–75 January effect and, 92–94 large-company stocks versus, 53–55, 75 “lottery ticket” premium and, 127 tracking error of, 75 Small investors, institutional investors versus, 59–61 Solnik, Bruno, 72 Sovereign risk, 72 S&P 500, 13, 38, 39, 55 as benchmark, 60, 78, 79, 80, 86, 88–89, 145 efficient frontier, 56–57 Spiders (SPDRS), 149 Spot rate, 135 Spread, 91, 92, 93, 96 Standard deviation, 5–8 defined, 6, 63 limitations of, 7 of manager returns, 96 in mean-variance analysis, 65 Standard error (SE), 87 Standard normal cumulative distribution function, 7 Stocks, Bonds, Bills, and Inflation (Ibbotson Associates), 9–10, 41–42, 178 Stocks for the Long Run (Siegel), 19, 136 Strategic asset allocation, 58–59 Survivorship bias, 101–102 Systematic risk, 13 t distribution function, 87 Tax-sheltered accounts: asset allocation for, 153–154 rebalancing, 108–109, 159–160 (See also Retirement accounts) 205 Taxable accounts: asset allocation for, 153–154 rebalancing, 160–161 Taxes: in asset allocation strategy, 145 capital gains capture, 102, 108 foreign tax credits, 161 market efficiency and, 102–103 Technological change: historical, impact of, 125 in new era of investing, 125 Templeton, John, 164 Thaler, Richard, 131, 142 Three-factor model (Fama and French), 120–124 Time horizon, 130, 131, 143–144, 167 Tracking error: defined, 75 determining tolerance for, 83, 145 of small-company stocks, 75 of various equity mixes, 79 Treasury bills: 1926–1998, 10–11 returns on, 25–26 as risk-free investments, 10, 15, 152 Treasury bonds: 1926–1998, 11–13, 42–45 ladders, 152 Treasury Inflation Protected Security (TIPS), 80, 131–132, 172 Treasury notes, 11 Turnover, 95, 102, 130–131, 145 Tweedy, Browne, 148–149, 162, 176 Utility functions, 7 Value averaging, 155–159 Value Averaging (Edleson), 176 Value index funds, 145 Value investing, 77, 111–124 defined, 118 growth investing versus, 117, 118–120 measures used in, 112–114 studies on, 115–118 three-factor model of, 120–124 Value premium, 121–123 206 Index VanEck Gold Fund, 21 Vanguard Group, 97–100, 146–148, 149, 150, 152, 156, 161–163 Variance, 7, 108–109 mean-variance analysis, 44–45, 64–71, 181–182 minimum-variance portfolios, 65–69 Variance drag, 69 Walz, Daniel T., 169 Websites, 178–180 Wilkinson, David, 56, 57, 181–182 Williams, John Burr, 127 Wilshire Associates, 120, 147, 162 World Equity Benchmark Securities (WEBS), 149–151 z values, 87 Zero correlation, 31 About the Author William Bernstein, Ph.D, M.D., is a practicing neurologist in Oregon.

Momentum in foreign and domestic equity asset classes exists, resulting in periodic asset overvaluation and undervaluation. Eventually long-term mean reversion occurs to correct these excesses. Over 2 decades ago, Eugene Fama made a powerful case that security price changes could not be predicted, and Burton Malkiel introduced the words “random walk” into the popular investing lexicon. Unfortunately, in a truly random-walk world, there is no advantage to portfolio rebalancing. If you rebalance, you profit only when the frogs in your portfolio turn into princes, and vice versa. In the real world, fortunately, there are subtle departures in random-walk behavior that the asset allocator-investor can exploit. Writer and money manager Ken Fisher calls this change in asset desirability, and the resultant short-term momentum and long-term mean reversion, the “Wall Street Waltz.”

Efficient-market theorists are fond of pointing out that there is no pattern to stock or market prices. (As we have already seen, this is not Odds and Ends 119 strictly true.) Growth-stock investors believe that they can pick those companies which will have persistent earnings growth and thus reap the benefits of their ever-increasing earnings stream. Unfortunately, established growth companies are very expensive, often selling at P/Es two or three times that of the market as a whole. A company growing 5% faster than the rest of the market and selling at a P/E twice the market’s will have to continue growing for another 14 years at that rate before the shareholder is fairly compensated. As we’ve already seen, stock price movements are essentially an unpredictable “random walk.” Interestingly, it turns out that earnings growth also exhibits random-walk behavior; a company with good earnings growth this year is quite likely to have poor earnings growth next year (and vice versa).


pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos

Benoit Mandelbrot, Black-Scholes formula, Brownian motion, business climate, business cycle, butter production in bangladesh, butterfly effect, capital asset pricing model, correlation coefficient, correlation does not imply causation, Daniel Kahneman / Amos Tversky, diversified portfolio, dogs of the Dow, Donald Trump, double entry bookkeeping, Elliott wave, endowment effect, Erdős number, Eugene Fama: efficient market hypothesis, four colour theorem, George Gilder, global village, greed is good, index fund, intangible asset, invisible hand, Isaac Newton, John Nash: game theory, Long Term Capital Management, loss aversion, Louis Bachelier, mandelbrot fractal, margin call, mental accounting, Myron Scholes, Nash equilibrium, Network effects, passive investing, Paul Erdős, Paul Samuelson, Ponzi scheme, price anchoring, Ralph Nelson Elliott, random walk, Richard Thaler, Robert Shiller, Robert Shiller, short selling, six sigma, Stephen Hawking, stocks for the long run, survivorship bias, transaction costs, ultimatum game, Vanguard fund, Yogi Berra

By starting with the basic up-down-up and down-up-down patterns of a stock’s possible movements, continually replacing each of these patterns’ three segments with smaller versions of one of the basic patterns chosen at random, and then altering the spikiness of the patterns to reflect changes in the stock’s volatility, Mandelbrot has constructed what he calls multifractal “forgeries.” The forgeries are patterns of price movement whose general look is indistinguishable from that of real stock price movements. In contrast, more conventional assumptions about price movements, say those of a strict random-walk theorist, lead to patterns that are noticeably different from real price movements. These multifractal patterns are so far merely descriptive, not predictive of specific price changes. In their modesty, as well as in their mathematical sophistication, they differ from the Elliott waves mentioned in chapter 3. Even this does not prove that chaos (in the mathematical sense) reigns in (part of) the market, but it is clearly a bit more than suggestive.

Weeding out some of the details, let’s assume for the sake of the argument (although Lo and MacKinlay don’t) that the thesis of Burton Malkiel’s classic book, A Random Walk Down Wall Street, is true and that the movement of the market as a whole is entirely random. Let’s also assume that each stock, when its fluctuations are examined in isolation, moves randomly. Given these assumptions it would nevertheless still be possible that the price movements of, say, 5 percent of stocks accurately predict the price movements of a different 5 percent of stocks one week later. The predictability comes from cross-correlations over time between stocks. (These associations needn’t be causal, but might merely be brute facts.) More concretely, let’s say stock X, when looked at in isolation, fluctuates randomly from week to week, as does stock Y. Yet if X’s price this week often predicts Y’s next week, this would be an exploitable opportunity and the strict random-walk hypothesis would be wrong.

Yet if X’s price this week often predicts Y’s next week, this would be an exploitable opportunity and the strict random-walk hypothesis would be wrong. Unless we delved deeply into such possible cross-correlations among stocks, all we would see would be a randomly fluctuating market populated by randomly fluctuating stocks. Of course, I’ve employed the typical mathematical gambit of considering an extreme case, but the example does suggest that there may be relatively simple elements of order in a market that appears to fluctuate randomly. There are other sorts of stock price anomalies that can lead to exploitable opportunities. Among the most well-known are so-called calendar effects whereby the prices of stocks, primarily small-firm stocks, rise disproportionately in January, especially during the first week of January. (The price of WCOM rose significantly in January 2001, and I was hoping this rise would repeat itself in January 2002.


pages: 206 words: 70,924

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

"Robert Solow", Albert Einstein, Bayesian statistics, Black-Scholes formula, Bretton Woods, Brownian motion, business cycle, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, David Ricardo: comparative advantage, discovery of penicillin, discrete time, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, floating exchange rates, full employment, Henri Poincaré, implied volatility, index fund, Isaac Newton, John Meriwether, John von Neumann, Joseph Schumpeter, Kenneth Arrow, Long Term Capital Management, Louis Bachelier, margin call, market clearing, martingale, means of production, moral hazard, Myron Scholes, Paul Samuelson, price stability, principal–agent problem, quantitative trading / quantitative finance, RAND corporation, random walk, risk tolerance, risk/return, Ronald Reagan, shareholder value, Sharpe ratio, short selling, stochastic process, Thales and the olive presses, Thales of Miletus, The Chicago School, the scientific method, too big to fail, transaction costs, tulip mania, Works Progress Administration, yield curve

Bachelier had already discovered this, though. His statement that stock prices could be modeled as a random walk according to a Weiner process was amenable to empirical verification. Alfred Cowles, who would found the Cowles Commission, and Herbert Jones explored and subsequently vindicated this notion that there is no memory effect in the price of stocks in a 1937 paper together.8 While the notion of the random walk has since been replaced with the less restrictive concept of a martingale process, much of finance pricing theory still retains the random walk because of its simple first and second moment characterization of price movements. The random walk of absolute prices Bachelier constructed a theory of absolute rather than relative price movements. Now we recognize it as some of the most analytically and theoretically complex work in stochastic calculus in finance until the late 1950s or 1960s.

Bachelier aimed to provide the theory behind these rather complex instruments that must be treated not only based on their prices but also on their expiration date, the exercise price, and the right to either buy (calls) or sell (puts) at that price. Bachelier modeled options pricing by noting how increments to the stock price would affect the price of the option derived from it. He assumed that the stock experienced identically and independently distributed random movements, which allowed him to use the central limit theorem to describe the probability distribution of these movements by the normal distribution. He also allowed a drift of zero mean of the security price and assumed that the variance of the price drift is proportional to the length of time of the random walk. In combination, he had described what we now call a Weiner process. The Times 105 While Bachelier was the first to apply Brownian motion to finance, the methodology is now commonplace.

Einstein and Bachelier both noted that, beyond a common drift element, the movement of a particle or a stock from one period to the next is uncorrelated. We now know this phenomenon as the random walk. We return to Bachelier’s model later in our discussion of options pricing theory, and more fully in the next volume of our series on Applications 33 the random walk and the efficient market hypothesis. Without fully anticipating the profound impact, he nonetheless created a wave of scientific innovation in finance. Others continued the tradition that he helped to establish. In Britain, the Financial Review of Reviews began analyzing the prices and volatility of various stocks and bonds. The precursors of the ratings agencies Standard & Poor’s, Fitch, and Moody’s began researching and analyzing the fundamental profitability of companies as a way of assuring investors that corporate bonds were sound.


pages: 425 words: 122,223

Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

"Robert Solow", Albert Einstein, asset allocation, backtesting, Benoit Mandelbrot, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, corporate raider, debt deflation, diversified portfolio, Eugene Fama: efficient market hypothesis, financial innovation, financial intermediation, fixed income, full employment, implied volatility, index arbitrage, index fund, interest rate swap, invisible hand, John von Neumann, Joseph Schumpeter, Kenneth Arrow, law of one price, linear programming, Louis Bachelier, mandelbrot fractal, martingale, means of production, money market fund, Myron Scholes, new economy, New Journalism, Paul Samuelson, profit maximization, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, stochastic process, Thales and the olive presses, the market place, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, Thorstein Veblen, transaction costs, transfer pricing, zero-coupon bond, zero-sum game

See also Capital Asset Pricing Model; Random price fluctuations; specific types of securities arbitrage Black/Scholes formula of: see Black/Scholes formula earnings ratio efficient markets and future of growth stocks information and interest rates and intrinsic value and manipulation risk and security analysis and shadow transfer trends value differentiation zero downside limit on “Price Movements in Speculative Markets: Trends or Random Walks” (Alexander) “Pricing of Options and Corporate Liabilities, The” (Black/Scholes) Probability theory Procter & Gamble Profit maximization Program trading Prospective yield “Proposal for a Smog Tax, A” (Sharpe) Puts: see Options Railroads RAND Random Character of Stock Prices, The (Cootner) “Random Difference Series for Use in the Analysis of Time Series, A” (Working) Random price fluctuations/random walks selection of securities and “Random Walks in Stock Market Prices” (Fama) Rational Expectations Hypothesis “Rational Theory of Warrant Pricing” (Samuelson) Regulation of markets Return analysis: see Risk/return ratios Review of Economics and Statistics Review of Economic Studies, The “RHM Warrant and Low-Price Stock Survey, The” Risk arbitrage calculations diversification and dominant expected return and minimalization portfolio premium return ratios Rosenberg’s model stock prices and of stocks vs. bonds systematic (beta) trade-offs valuation of assets and “Risk and the Evaluation of Pension Fund Performance” (Fama) Risk-free assets Rosenberg Institutional Equity Management (RIEM) “Safety First and the Holding of Assets” (Roy) Samsonite Savings rates Scott Paper Securities analysis Securities and Exchange Commission Security Analysis (Graham/Dodd) Security selection Separation Theorem Shadow prices “Simplified Model for Portfolio Analysis, A” (Sharpe) Singer Manufacturing Company Single-index model Sloan School of Management Standard & Poor’s 500 index “State of the Art in Our Profession, The” (Vertin) Stock(s) cash ratios common expected return on growth income international legal restrictions on market value variance volatility Stock market (general discussion) Black Monday (October, 1987, crash) “Stock Market ‘Patterns’ and Financial Analysis” (Roberts) Supply and demand theory Swaps Tactical asset allocation theory Tampax Taxes.

He then throws down the gauntlet to the chartists and technical analysts who believe that the past pattern of stock prices makes future prices predictable: The chartist must admit that the evidence in favor of the random walk model is both consistent and voluminous, whereas there is precious little published in discussion of rigorous empirical tests of the various technical theories. If the chartist rejects the evidence of the random walk model, his position is weak if his own theories have not been subjected to equally rigorous tests. This, I believe, is the challenge that the random walk theory makes.14 Now Fama moves to a deeper matter: the uses and value of information itself. This is a more serious issue than the accomplishments of the technical analysts, who eschew information and believe that market prices tell their own story of what is to come. It is also an intensely controversial issue.

The mathematical formula that describes this phenomenon was one of Bachelier’s crowning achievements. Over time, in the literature on finance, Brownian motion came to be called the random walk, which someone once described as the path a drunk might follow at night in the light of a lamppost. No one knows who first used this expression, but it became increasingly familiar among academics during the 1960s, much to the annoyance of financial practitioners. Eugene Fama of the University of Chicago, one of the first and most enthusiastic proponents of the concept, tells me that random walk “is an ancient statistical term; nobody alive can claim it.”13 In later years, the primary focus of research on capital markets was on determining whether or not the random walk is a valid description of security price movements. Bachelier himself, hardly a modest man, ended his dissertation with this flat statement: “It is evident that the present theory resolves the majority of problems in the study of speculation by the calculus of probability.”14 Despite its importance, Bachelier’s thesis was lost until it was rediscovered quite by accident in the 1950s by Jimmie Savage, a mathematical statistician at Chicago.


pages: 319 words: 106,772

Irrational Exuberance: With a New Preface by the Author by Robert J. Shiller

Andrei Shleifer, asset allocation, banking crisis, Benoit Mandelbrot, business cycle, buy and hold, computer age, correlation does not imply causation, Daniel Kahneman / Amos Tversky, demographic transition, diversification, diversified portfolio, equity premium, Everybody Ought to Be Rich, experimental subject, hindsight bias, income per capita, index fund, Intergovernmental Panel on Climate Change (IPCC), Joseph Schumpeter, Long Term Capital Management, loss aversion, mandelbrot fractal, market bubble, market design, market fundamentalism, Mexican peso crisis / tequila crisis, Milgram experiment, money market fund, moral hazard, new economy, open economy, pattern recognition, Ponzi scheme, price anchoring, random walk, Richard Thaler, risk tolerance, Robert Shiller, Robert Shiller, Ronald Reagan, Small Order Execution System, spice trade, statistical model, stocks for the long run, survivorship bias, the market place, Tobin tax, transaction costs, tulip mania, urban decay, Y2K

., Scottsdale, Arizona, and Roosevelt, New Jersey The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (R1997) (Permanence of Paper) http://pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To Ben and Derek This page intentionally left blank Contents List of Figures and Tables Preface Acknowledgments One ix xi xix The Stock Market Level in Historical Perspective 3 Part One Structural Factors Two Three Precipitating Factors: The Internet, the Baby Boom, and Other Events Amplification Mechanisms: Naturally Occurring Ponzi Processes 17 44 Part Two Cultural Factors Four Five Six The News Media New Era Economic Thinking New Eras and Bubbles around the World vii 71 96 118 viii C ONT ENTS Part Three Psychological Factors Seven Eight Psychological Anchors for the Market Herd Behavior and Epidemics 135 148 Part Four Attempts to Rationalize Exuberance Nine Ten Efficient Markets, Random Walks, and Bubbles Investor Learning—and Unlearning 171 191 Part Five A Call to Action Eleven Speculative Volatility in a Free Society Notes References Index 235 269 283 203 Figures and Tables Figures 1.1 1.2 1.3 9.1 Stock Prices and Earnings, 1871–2000 Price-Earnings Ratio, 1881–2000 Price-Earnings Ratio as Predictor of Ten-Year Returns Stock Price and Dividend Present Value, 1871–2000 6 8 11 186 Tables 6.1 6.2 6.3 6.4 Largest Recent One-Year Real Stock Price Index Increases Largest Recent One-Year Real Stock Price Index Decreases Largest Recent Five-Year Real Stock Price Index Increases Largest Recent Five-Year Real Stock Price Index Decreases ix 119 120 121 122 This page intentionally left blank Preface T his book is a broad study, drawing on a wide range of published research and historical evidence, of the enormous recent stock market boom.

Part Four Attempts to Rationalize Exuberance This page intentionally left blank Nine Efficient Markets, Random Walks, and Bubbles T he theory that financial markets are very efficient, and the extensive research investigating this theory, form the leading intellectual basis for arguments against the idea that markets are vulnerable to excessive exuberance or bubbles. The efficient markets theory asserts that all financial prices accurately reflect all public information at all times. In other words, financial assets are always priced correctly, given what is publicly known, at all times. Price may appear to be too high or too low at times, but, according to the efficient markets theory, this appearance must be an illusion. Stock prices, by this theory, approximately describe “random walks” through time: the price changes are unpredictable since they occur only in response to genuinely new information, which by the very fact that is new is unpredictable.

One method for judging whether there is evidence in support of the basic validity of the efficient markets theory, which I published in an article in the American Economic Review in 1981 (at the same time as a similar paper by Stephen LeRoy and Richard Porter appeared), is to see whether the very volatility of speculative prices, such as stock prices, can be justified by the variability of dividends over long intervals of time. If the stock price move- E F F ICIE N T MARKE TS , RANDOM WALKS, AND BUBB LES 185 ments are to be justified in terms of the future dividends that firms pay out, as the basic version of the efficient markets theory would imply, then under efficient markets we cannot have volatile prices without subsequently volatile dividends.23 In fact, my article concluded, no movement of U.S. aggregate stock prices beyond the trend growth of prices has ever been subsequently justified by dividend movements, as the dividend present value has shown an extraordinarily smooth growth path.


pages: 752 words: 131,533

Python for Data Analysis by Wes McKinney

backtesting, cognitive dissonance, crowdsourcing, Debian, Firefox, Google Chrome, Guido van Rossum, index card, random walk, recommendation engine, revision control, sentiment analysis, Sharpe ratio, side project, sorting algorithm, statistical model, type inference

Partial list of numpy.random functions FunctionDescription seed Seed the random number generator permutation Return a random permutation of a sequence, or return a permuted range shuffle Randomly permute a sequence in place rand Draw samples from a uniform distribution randint Draw random integers from a given low-to-high range randn Draw samples from a normal distribution with mean 0 and standard deviation 1 (MATLAB-like interface) binomial Draw samples a binomial distribution normal Draw samples from a normal (Gaussian) distribution beta Draw samples from a beta distribution chisquare Draw samples from a chi-square distribution gamma Draw samples from a gamma distribution uniform Draw samples from a uniform [0, 1) distribution Example: Random Walks An illustrative application of utilizing array operations is in the simulation of random walks. Let’s first consider a simple random walk starting at 0 with steps of 1 and -1 occurring with equal probability. A pure Python way to implement a single random walk with 1,000 steps using the built-in random module: import random position = 0 walk = [position] steps = 1000 for i in xrange(steps): step = 1 if random.randint(0, 1) else -1 position += step walk.append(position) See Figure 4-4 for an example plot of the first 100 values on one of these random walks. Figure 4-4. A simple random walk You might make the observation that walk is simply the cumulative sum of the random steps and could be evaluated as an array expression.

First, I’ll use scaled prices for the SPY exchange-traded fund as a proxy for the S&P 500 index: In [127]: import pandas.io.data as web # Approximate price of S&P 500 index In [128]: px = web.get_data_yahoo('SPY')['Adj Close'] * 10 In [129]: px Out[129]: Date 2011-08-01 1261.0 2011-08-02 1228.8 2011-08-03 1235.5 ... 2012-07-25 1339.6 2012-07-26 1361.7 2012-07-27 1386.8 Name: Adj Close, Length: 251 Now, a little bit of setup. I put a couple of S&P 500 future contracts and expiry dates in a Series: from datetime import datetime expiry = {'ESU2': datetime(2012, 9, 21), 'ESZ2': datetime(2012, 12, 21)} expiry = Series(expiry).order() expiry then looks like: In [131]: expiry Out[131]: ESU2 2012-09-21 00:00:00 ESZ2 2012-12-21 00:00:00 Then, I use the Yahoo! Finance prices along with a random walk and some noise to simulate the two contracts into the future: np.random.seed(12347) N = 200 walk = (np.random.randint(0, 200, size=N) - 100) * 0.25 perturb = (np.random.randint(0, 20, size=N) - 10) * 0.25 walk = walk.cumsum() rng = pd.date_range(px.index[0], periods=len(px) + N, freq='B') near = np.concatenate([px.values, px.values[-1] + walk]) far = np.concatenate([px.values, px.values[-1] + walk + perturb]) prices = DataFrame({'ESU2': near, 'ESZ2': far}, index=rng) prices then has two time series for the contracts that differ from each other by a random amount: In [133]: prices.tail() Out[133]: ESU2 ESZ2 2013-04-16 1416.05 1417.80 2013-04-17 1402.30 1404.55 2013-04-18 1410.30 1412.05 2013-04-19 1426.80 1426.05 2013-04-22 1406.80 1404.55 One way to splice time series together into a single continuous series is to construct a weighting matrix.

Thus, I use the np.random module to draw 1,000 coin flips at once, set these to 1 and -1, and compute the cumulative sum: In [215]: nsteps = 1000 In [216]: draws = np.random.randint(0, 2, size=nsteps) In [217]: steps = np.where(draws > 0, 1, -1) In [218]: walk = steps.cumsum() From this we can begin to extract statistics like the minimum and maximum value along the walk’s trajectory: In [219]: walk.min() In [220]: walk.max() Out[219]: -3 Out[220]: 31 A more complicated statistic is the first crossing time, the step at which the random walk reaches a particular value. Here we might want to know how long it took the random walk to get at least 10 steps away from the origin 0 in either direction. np.abs(walk) >= 10 gives us a boolean array indicating where the walk has reached or exceeded 10, but we want the index of the first 10 or -10. Turns out this can be computed using argmax, which returns the first index of the maximum value in the boolean array (True is the maximum value): In [221]: (np.abs(walk) >= 10).argmax() Out[221]: 37 Note that using argmax here is not always efficient because it always makes a full scan of the array. In this special case once a True is observed we know it to be the maximum value. Simulating Many Random Walks at Once If your goal was to simulate many random walks, say 5,000 of them, you can generate all of the random walks with minor modifications to the above code.


pages: 517 words: 139,477

Stocks for the Long Run 5/E: the Definitive Guide to Financial Market Returns & Long-Term Investment Strategies by Jeremy Siegel

Asian financial crisis, asset allocation, backtesting, banking crisis, Black-Scholes formula, break the buck, Bretton Woods, business cycle, buy and hold, buy low sell high, California gold rush, capital asset pricing model, carried interest, central bank independence, cognitive dissonance, compound rate of return, computer age, computerized trading, corporate governance, correlation coefficient, Credit Default Swap, Daniel Kahneman / Amos Tversky, Deng Xiaoping, discounted cash flows, diversification, diversified portfolio, dividend-yielding stocks, dogs of the Dow, equity premium, Eugene Fama: efficient market hypothesis, eurozone crisis, Everybody Ought to Be Rich, Financial Instability Hypothesis, fixed income, Flash crash, forward guidance, fundamental attribution error, housing crisis, Hyman Minsky, implied volatility, income inequality, index arbitrage, index fund, indoor plumbing, inflation targeting, invention of the printing press, Isaac Newton, joint-stock company, London Interbank Offered Rate, Long Term Capital Management, loss aversion, market bubble, mental accounting, money market fund, mortgage debt, Myron Scholes, new economy, Northern Rock, oil shock, passive investing, Paul Samuelson, Peter Thiel, Ponzi scheme, prediction markets, price anchoring, price stability, purchasing power parity, quantitative easing, random walk, Richard Thaler, risk tolerance, risk/return, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, stocks for the long run, survivorship bias, technology bubble, The Great Moderation, the payments system, The Wisdom of Crowds, transaction costs, tulip mania, Tyler Cowen: Great Stagnation, Vanguard fund

In 1965, Professor Paul Samuelson of MIT showed that the randomness in security prices did not contradict the laws of supply and demand.4 In fact, such randomness was a result of a free and efficient market in which investors had already incorporated all the known factors influencing the price of the stock. This is the crux of the efficient market hypothesis. If the market is efficient, prices will change only when new, unanticipated information is released to the market. Since unanticipated information is as likely to be better than expected as it is to be worse than expected, the resulting movement in stock prices is random. Price charts will therefore look like a random walk and cannot be predicted.5 SIMULATIONS OF RANDOM STOCK PRICES If stock prices are indeed random, their movements should not be distinguishable from simulations generated randomly by a computer. Figure 20-1 extends the experiment conceived by Professor Roberts 60 years ago. Instead of generating only closing prices, I programmed the computer to generate intraday prices, creating the popular high-low-close bar graphs that are found in most newspapers and chart publications.

The dashed bars in Figure 6-2 show the decline in risk predicted under the random walk assumption. But the historical data show that the random walk hypothesis cannot be maintained for equities. This occurs since the actual risk of average stock returns declines far faster than predicted by the random walk hypothesis because of the mean reversion of equity returns. The standard deviation of average returns for fixed-income assets, on the other hand, does not fall as fast as the random walk theory predicts. This is a manifestation of mean aversion of bond returns. Mean aversion implies that once an asset’s return deviates from its long-run average, there is an increased chance that it will deviate further, rather than return to more normal levels. Mean aversion of bond returns is especially characteristic of hyperinflations, where price changes proceed at an accelerating pace, rendering paper assets worthless.

Given these striking differences, it might seem puzzling that the holding period has almost never been considered in standard portfolio theory. This is because modern portfolio theory was established when the vast majority of the academic profession supported the random walk theory of security prices. As noted earlier, when prices are a random walk, the risk over any holding period is a simple function of the risk over a single period, so that the relative risk of different asset classes does not depend on the holding period. In that case the efficient frontier is invariant to the time period, and asset allocation does not depend on the investment horizon of the investor. When security markets do not obey random walks, that conclusion cannot be maintained.6 CONCLUSION No one denies that, in the short run, stocks are riskier than fixed-income assets. But in the long run, history has shown that stocks are actually safer than bonds for long-term investors whose goal is to preserve the purchasing power of their wealth.


Analysis of Financial Time Series by Ruey S. Tsay

Asian financial crisis, asset allocation, Bayesian statistics, Black-Scholes formula, Brownian motion, business cycle, capital asset pricing model, compound rate of return, correlation coefficient, data acquisition, discrete time, frictionless, frictionless market, implied volatility, index arbitrage, Long Term Capital Management, market microstructure, martingale, p-value, pattern recognition, random walk, risk tolerance, short selling, statistical model, stochastic process, stochastic volatility, telemarketer, transaction costs, value at risk, volatility smile, Wiener process, yield curve

In some studies, interest rates, foreign exchange rates, or the price series of an asset are of interest. These series tend to be nonstationary. For a price series, the nonstationarity is mainly due to the fact that there is no fixed level for the price. In the time series literature, such a nonstationary series is called unit-root nonstationary time series. The best known example of unit-root nonstationary time series is the random-walk model. 2.7.1 Random Walk A time series { pt } is a random walk if it satisfies pt = pt−1 + at , (2.32) where p0 is a real number denoting the starting value of the process and {at } is a white noise series. If pt is the log price of a particular stock at date t, then p0 could be the log price of the stock at its initial public offering (i.e., the logged IPO price). If at has a symmetric distribution around zero, then conditional on pt−1 , pt has a 50–50 chance to go up or down, implying that pt would go up or down at random.

If at has a symmetric distribution around zero, then conditional on pt−1 , pt has a 50–50 chance to go up or down, implying that pt would go up or down at random. If we treat the random-walk model as a special AR(1) model, then the coefficient of pt−1 is unity, which does not satisfy the weak stationarity condition of an AR(1) model. A random-walk series is, therefore, not weakly stationary, and we call it a unit-root nonstationary time series. The random-walk model has been widely considered as a statistical model for the movement of logged stock prices. Under such a model, the stock price is not predictable or mean reverting. To see this, the 1-step ahead forecast of model (2.32) at the forecast origin h is p̂h (1) = E( ph+1 | ph , ph−1 , . . .) = ph , which is the log price of the stock at the forecast origin. Such a forecast has no practical value. The 2-step ahead forecast is UNIT- ROOT NONSTATIONARITY 57 p̂h (2) = E( ph+2 | ph , ph−1 , . . .) = E( ph+1 + ah+2 | ph , ph−1 , . . .) = E( ph+1 | ph , ph−1 , . . .) = p̂h (1) = ph , which again is the log price at the forecast origin.

Theoretically, this means that pt can assume any real value for a sufficiently large t. For the log price pt of an individual stock, this is plausible. Yet for market indexes, negative log price is very rare if it happens at all. In this sense, the adequacy of a random-walk model for market indexes is questionable. Third, from the representation, ψi = 1 for all i. Thus, the impact of any past shock at−i on pt does not decay over time. Consequently, the series has a strong memory as it remembers all of the past shocks. In economics, the shocks are said to have a permanent effect on the series. 2.7.2 Random Walk with a Drift As shown by empirical examples considered so far, the log return series of a market index tends to have a small and positive mean. This implies that the model for the log price is pt = µ + pt−1 + at , (2.33) where µ = E( pt − pt−1 ) and {at } is a white noise series.


pages: 374 words: 114,600

The Quants by Scott Patterson

Albert Einstein, asset allocation, automated trading system, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Bernie Sanders, Black Swan, Black-Scholes formula, Blythe Masters, Bonfire of the Vanities, Brownian motion, buttonwood tree, buy and hold, buy low sell high, capital asset pricing model, centralized clearinghouse, Claude Shannon: information theory, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Donald Trump, Doomsday Clock, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Gordon Gekko, greed is good, Haight Ashbury, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, index fund, invention of the telegraph, invisible hand, Isaac Newton, job automation, John Meriwether, John Nash: game theory, Kickstarter, law of one price, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, merger arbitrage, money market fund, Myron Scholes, NetJets, new economy, offshore financial centre, old-boy network, Paul Lévy, Paul Samuelson, Ponzi scheme, quantitative hedge fund, quantitative trading / quantitative finance, race to the bottom, random walk, Renaissance Technologies, risk-adjusted returns, Robert Mercer, Rod Stewart played at Stephen Schwarzman birthday party, Ronald Reagan, Sergey Aleynikov, short selling, South Sea Bubble, speech recognition, statistical arbitrage, The Chicago School, The Great Moderation, The Predators' Ball, too big to fail, transaction costs, value at risk, volatility smile, yield curve, éminence grise

It would also give birth to its own destructive forces and pave the way to a series of financial catastrophes, culminating in an earthshaking collapse that erupted in August 2007. Like Thorp’s methodology for pricing warrants, an essential component of the Black-Scholes formula was the assumption that stocks moved in a random walk. Stocks, in other words, are assumed to move in antlike zigzag patterns just like the pollen particles observed by Brown in 1827. In their 1973 paper, Black and Scholes wrote that they assumed that the “stock price follows a random walk in continuous time.” Just as Thorp had already discovered, this allowed investors to determine the relevant probabilities for volatility—how high or low a stock or option would move in a certain time frame. Hence, the theory that had begun with Robert Brown’s scrutiny of plants, then led to Bachelier’s observations about bond prices, finally reached a most pragmatic conclusion—a formula that Wall Street would use to trade billions of dollars’ worth of stock and options.

“It is not easy to get rich in Las Vegas, at Churchill Downs, or at the local Merrill Lynch office,” he wrote. At the time, Samuelson was becoming an éminence grise of the economic community. If he thought the market followed a random walk, that meant everyone had to get on board or have a damn good reason not to. Most agreed, including one of Samuelson’s star students, Robert Merton, one of the co-creators of the Black-Scholes option-pricing formula. Another acolyte was Burton Malkiel, who went on to write A Random Walk Down Wall Street. It was Fama, however, who connected all of the dots and put the efficient-market hypothesis on the map as a central feature of modern portfolio theory. The idea that the market is an efficient, randomly churning price-processing machine has many odd consequences. Fama postulates a vast, swarming world of investors constantly searching for inefficiencies—those hungry piranhas circling in wait of fresh meat.

The future movement of a stock—a variable known to quants as volatility—is random, and therefore quantifiable. And if the warrant is priced in a way that underestimates, or overestimates, its likely volatility, money can be made. Discovering how to price volatility was the key to unlocking the stock warrant treasure trove. Say you own a warrant for IBM. The current value of IBM’s stock is $100. The warrant, which expires in twelve months, will be valuable only if IBM is worth $110 at some point during that twelve-month period. If you can determine how volatile IBM’s stock is—how likely it is that it will hit $110 during that time period—you then know how much the warrant is worth. Thorp discovered that by plugging in the formula for Brownian motion, the random walk model, in addition to an extra variable for whether the stock itself tends to rise more or less than other stocks, he could know better than almost anyone else in the market what the IBM warrant was worth.


pages: 1,088 words: 228,743

Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

Andrei Shleifer, asset allocation, asset-backed security, availability heuristic, backtesting, balance sheet recession, bank run, banking crisis, barriers to entry, Bernie Madoff, Black Swan, Bretton Woods, business cycle, buy and hold, buy low sell high, capital asset pricing model, capital controls, Carmen Reinhart, central bank independence, collateralized debt obligation, commoditize, commodity trading advisor, corporate governance, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, debt deflation, deglobalization, delta neutral, demand response, discounted cash flows, disintermediation, diversification, diversified portfolio, dividend-yielding stocks, equity premium, Eugene Fama: efficient market hypothesis, fiat currency, financial deregulation, financial innovation, financial intermediation, fixed income, Flash crash, framing effect, frictionless, frictionless market, G4S, George Akerlof, global reserve currency, Google Earth, high net worth, hindsight bias, Hyman Minsky, implied volatility, income inequality, incomplete markets, index fund, inflation targeting, information asymmetry, interest rate swap, invisible hand, Kenneth Rogoff, laissez-faire capitalism, law of one price, London Interbank Offered Rate, Long Term Capital Management, loss aversion, margin call, market bubble, market clearing, market friction, market fundamentalism, market microstructure, mental accounting, merger arbitrage, mittelstand, moral hazard, Myron Scholes, negative equity, New Journalism, oil shock, p-value, passive investing, Paul Samuelson, performance metric, Ponzi scheme, prediction markets, price anchoring, price stability, principal–agent problem, private sector deleveraging, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, random walk, reserve currency, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, riskless arbitrage, Robert Shiller, Robert Shiller, savings glut, selection bias, Sharpe ratio, short selling, sovereign wealth fund, statistical arbitrage, statistical model, stochastic volatility, stocks for the long run, survivorship bias, systematic trading, The Great Moderation, The Myth of the Rational Market, too big to fail, transaction costs, tulip mania, value at risk, volatility arbitrage, volatility smile, working-age population, Y2K, yield curve, zero-coupon bond, zero-sum game

The model assumes that earnings streams follow a random walk; yet investors mistakenly believe that observed earnings come from either of two regimes—a normal mean-reverting regime or a momentum regime—and they switch back and forth between the two mindsets. They try to infer the prevailing regime from the data, predicting earnings reversals when they believe that normal conditions apply, but extrapolating apparent earnings trends when they observe a string of same sign shocks and infer that a momentum regime is more likely. The first type of mistake (proxying for conservatism) leads to price underreactions and momentum, while the second type of mistake leads to (delayed) price overreactions. Because the true earnings process is a random walk, the extrapolating investor will eventually be disappointed, resulting in long-term price reversal and value effects [8]

Adding the evidence that estimated ex ante bond risk premia track expected inflation, it seems fair to assume that inflation uncertainty and the IRP are both level dependent. Inflation persistence rose with inflation level until 1980. During the gold standard, prices could go persistently up or down but the best long-term forecasts used to be for no change. That is, the price level followed a random walk and recent inflation had no ability to predict future inflation. Between the 1950s and the 1970s, as inflation’s persistence gradually rose, the best time series forecast of future inflation shifted from zero to the most recent inflation rate. That is, instead of the price level following a random walk, the inflation rate did so (the statistical persistence parameter rose from 0 to 1, before reversing after 1980). At least the process did not become more explosive in developed markets, as it does when hyperinflation psychology takes hold.

Antti Ilmanen Bad Homburg, November 2010 Abbreviations and acronyms AM Arithmetic Mean ATM At The Money (option) AUM Assets Under Management BEI Break-Even Inflation BF Behavioral Finance B/P Book/Price, book-to-market ratio BRP Bond Risk Premium, term premium B-S Black–Scholes C-P BRP Cochrane–Piazzesi Bond Risk Premium CAPM Capital Asset Pricing Model CAY Consumption wealth ratio CB Central Bank CCW Covered Call Writing CDO Collateralized Debt Obligation CDS Credit Default Swap CF Cash Flow CFNAI Chicago Fed National Activity Index CFO Chief Financial Officer CMD Commodity (futures) CPIyoy Consumer Price Inflation year on year CRB Commodity Research Bureau CRP Credit Risk Premium (over Treasury bond) CRRA Constant Relative Risk Aversion CTA Commodity Trading Advisor DDM Dividend Discount Model DJ CS Dow Jones Credit Suisse DMS Dimson–Marsh–Staunton D/P Dividend/Price (ratio), dividend yield DR Diversification Return E( ) Expected (conditional expectation) EMH Efficient Markets Hypothesis E/P Earnings/Price ratio, earnings yield EPS Earnings Per Share ERP Equity Risk Premium ERPB Equity Risk Premium over Bond (Treasury) ERPC Equity Risk Premium over Cash (Treasury bill) F Forward price or futures price FF Fama–French FI Fixed Income FoF Fund of Funds FX Foreign eXchange G Growth rate GARCH Generalized AutoRegressive Conditional Heteroskedasticity GC General Collateral repo rate (money market interest rate) GDP Gross Domestic Product GM Geometric Mean, also compound annual return GP General Partner GSCI Goldman Sachs Commodity Index H Holding-period return HF Hedge Fund HFR Hedge Fund Research HML High Minus Low, a value measure, also VMG HNWI High Net Worth Individual HPA House Price Appreciation (rate) HY High Yield, speculative-rated debt IG Investment Grade (rated debt) ILLIQ Measure of a stock’s illiquidity: average absolute daily return over a month divided by dollar volume IPO Initial Public Offering IR Information Ratio IRP Inflation Risk Premium ISM Business confidence index ITM In The Money (option) JGB Japanese Government Bond K-W BRP Kim–Wright Bond Risk Premium LIBOR London InterBank Offered Rate, a popular bank deposit rate LP Limited Partner LSV Lakonishok–Shleifer–Vishny LtA Limits to Arbitrage LTCM Long-Term Capital Management MA Moving Average MBS (fixed rate, residential) Mortgage-Backed Securities MIT-CRE MIT Center for Real Estate MOM Equity MOMentum proxy MSCI Morgan Stanley Capital International MU Marginal Utility NBER National Bureau of Economic Research NCREIF National Council of Real Estate Investment Fiduciaries OAS Option-Adjusted (credit) Spread OTM Out of The Money (option) P Price P/B Price/Book (valuation ratio) P/E Price/Earnings (valuation ratio) PE Private Equity PEH Pure Expectations Hypothesis PT Prospect Theory r Excess return R Real (rate) RE Real Estate REITs Real Estate Investment Trusts RWH Random Walk Hypothesis S Spot price, spot rate SBRP Survey-based Bond Risk Premium SDF Stochastic Discount Factor SMB Small Minus Big, size premium proxy SR Sharpe Ratio SWF Sovereign Wealth Fund TED Treasury–Eurodollar (deposit) rate spread in money markets TIPS Treasury Inflation-Protected Securities, real bonds UIP Uncovered Interest Parity (hypothesis) VaR Value at Risk VC Venture Capital VIX A popular measure of the implied volatility of S&P 500 index options VMG Value Minus Growth, equity value premium proxy WDRA Wealth-Dependent Risk Aversion X Cash flow Y Yield YC Yield Curve (steepness), term spread YTM Yield To Maturity YTW Yield To Worst Disclaimer Antti Ilmanen is a Senior Portfolio Manager at Brevan Howard, one of Europe’s largest hedge fund managers.


pages: 320 words: 33,385

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

asset allocation, backtesting, barriers to entry, Brownian motion, capital asset pricing model, constrained optimization, credit crunch, Credit Default Swap, discounted cash flows, discrete time, diversification, diversified portfolio, en.wikipedia.org, fixed income, implied volatility, interest rate swap, market friction, market microstructure, p-value, performance metric, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, Thomas Bayes, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-sum game

In Section II.5.3.7 we prove that the discrete time version of (I.3.142) is a stationary AR(1) model. I.3.7.3 Stochastic Models for Asset Prices and Returns Time series of asset prices behave quite differently from time series of returns. In efficient markets a time series of prices or log prices will follow a random walk. More generally, even in the presence of market frictions and inefficiencies, prices and log prices of tradable assets are integrated stochastic processes. These are fundamentally different from the associated returns, which are generated by stationary stochastic processes. Figures I.3.28 and I.3.29 illustrate the fact that prices and returns are generated by very different types of stochastic process. Figure I.3.28 shows time series of daily prices (lefthand scale) and log prices (right-hand scale) of the Dow Jones Industrial Average (DJIA) DJIA 12000 9.4 Log DJIA 9.3 11000 9.2 10000 9.1 9000 9 8000 8.9 Sep-01 May-01 Jan-01 Sep-00 May-00 Jan-00 Sep-99 May-99 Jan-99 Sep-98 May-98 8.8 Jan-98 7000 Figure I.3.28 Daily prices and log prices of DJIA index 56 This is not the only possible discretization of a continuous increment.

Probability and Statistics 139 Application of Itô’s lemma with f = ln S shows that a continuous time representation of geometric Brownian motion that is equivalent to the geometric Brownian motion (I.3.143) but is translated into a process for log prices is the arithmetic Brownian motion, d ln St = − 21 2 dt + dWt (I.3.145) We already know what a discretization of (I.3.145) looks like. The change in the log price is the log return, so using the standard discrete time notation Pt for a price at time t we have d ln St → ln Pt Hence the discrete time equivalent of (I.3.145) is ln Pt = + $t where = prices, i.e. − 1 2 2 $t ∼ NID 0 2 (I.3.146) . This is equivalent to a discrete time random walk model for the log ln Pt = + ln Pt−1 + $t $t ∼ NID 0 2 (I.3.147) To summarize, the assumption of geometric Brownian motion for prices in continuous time is equivalent to the assumption of a random walk for log prices in discrete time. I.3.7.4 Jumps and the Poisson Process A Poisson process, introduced in Section I.3.3.2, is a stochastic process governing the occurrences of events through time.

. • Continuous time stochastic processes are represented as stochastic differential equations (SDEs). The most famous example of an SDE in finance is geometric Brownian motion. This is introduced below, but its application to option pricing is not discussed until Chapter III.3. The first two subsections define what is meant by a stationary or ‘mean-reverting’ stochastic process in discrete and continuous time. We contrast this with a particular type of nonstationary process which is called a ‘random walk’. Then Section I.3.7.3 focuses on some standard discrete and continuous time models for the evolution of financial asset prices and returns. The most basic assumption in both types of models is that the prices of traded assets follow a random walk, and from this it follows that their returns follow a stationary process. I.3.7.1 Stationary and Integrated Processes in Discrete Time This section introduces the time series models that are used to model stationary and integrated processes in discrete time.


Capital Ideas Evolving by Peter L. Bernstein

Albert Einstein, algorithmic trading, Andrei Shleifer, asset allocation, business cycle, buy and hold, buy low sell high, capital asset pricing model, commodity trading advisor, computerized trading, creative destruction, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, diversification, diversified portfolio, endowment effect, equity premium, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, high net worth, hiring and firing, index fund, invisible hand, Isaac Newton, John Meriwether, John von Neumann, Joseph Schumpeter, Kenneth Arrow, London Interbank Offered Rate, Long Term Capital Management, loss aversion, Louis Bachelier, market bubble, mental accounting, money market fund, Myron Scholes, paper trading, passive investing, Paul Samuelson, price anchoring, price stability, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical model, survivorship bias, systematic trading, technology bubble, The Wealth of Nations by Adam Smith, transaction costs, yield curve, Yogi Berra, zero-sum game

So long as a minute minority of investors, possessed of considerable assets, can seek gain by trading against willful uninformed bettors, then Limited Efficiency of Markets will be empirically observable. The temporary appearance of aberrant price profiles coaxes action from alert traders who act gleefully to wipe out the aberration.” In more colorful language, he has made the same point this way: “My pitch on this occasion is not exclusively or even primarily aimed at practical men. The less of them who become sophisticated, the better for us happy few.”5 bern_c03.qxd 3/23/07 9:01 AM Page 41 Paul A. Samuelson 41 The consequence of all this market activity is a more complex state of affairs than we would find in a truly random walk.* As Samuelson points out, “After numerous people carefully weigh new information arriving about the future, all that is pragmatically knowable is already in current pricing patterns. This makes speculative prices behave like what mathematicians called a ‘martingale,’ where in the next period prices may as likely change more than the total market index or change less.”

., 49 Micro-Cap Equity (1999), 16, 17, 18 Micro-eff iciency hypothesis, 72–76 Mid-Cap Growth Equity (1992), 16–17 Miller, Bill, 170 Miller, Merton, 30, 239 Miller-farmer risk transfer, 115–117 Mishkin, Frederic, 240 Modern Portfolio, 154 –163 Modern Portfolio Theory, 30, 154 Modigliani, Franco inf luence on BGI by, 143 inf luence on Lo by, 59 on irrational pricing, 29 Nobel Prize awarded to, 239 Shiller’s work with, 65, 66 bern_z04bindex.qxd 4/3/07 8:20 AM Page 275 Index Modigliani-Miller corporate f inance approach, 67 Momentum risk, 27 Monte Carlo simulations, 97, 162 Morgan Stanley, 196, 201 Morgan Stanley Capital International (MSCI), 242 Morningstar, 20 Multifactor models, 143–144, 171 Mutual funds Harlow-Brown paradigm on performance of, 22–23 Stagecoach Fund, 128, 132–133 underperformance of, 21–22 Mutual funds performance disappointing Stagecoach, 128, 133 Harlow-Brown paradigm on, 22–23 underperformance of, 21–22 Nagel, Stegan, 29–30 NASDAQ anticipated reduction in cost of trading on, 54 bubble (1998–2000) of, 29–30 Neoclassical theory equilibrium models of, 100 Merton’s on valid predictions of, 56, 57 Ross on, 26 See also Capital Ideas The New Financial Order (Shiller), 81, 82 New York Stock Exchange anticipated reduction in cost of trading on, 54 high rate of turnover on the, 56 Nikko Investment Advisors (Tokyo), 136 No-arbitrage, 26 275 Noise traders, 19–20 A Non-Random Walk Down Wall Street ( Lo and MacKinlay), 62, 70 Non-stationarity, 71–72 “A Note on Measurement of Utility” (Samuelson), 39 NYCERS (New York City Employees Retirement System), 182 Odean, Terrance, 13 Ohm’s Law, 112 Omega risk transfer described, 111–112 Platinum Grove’s use of, 111–112, 118 Optimizers Black-Litterman model on choosing constraints for, 229 Black-Litterman solution to mean/variance, 225–228 Markowitz’s risk and return trade-off, 166–167 Options pricing theory, 238 The Origin of Species ( Darwin), 62 Outcome-investing, 97 Passive alphas, 206–209 Pensions & Investments (trade paper), 107–108 Perold, André, 168–169 PGGM (Stichting Pensionfonds) 241 Pimco Investment Management (California) alpha engine strategy used by, 185–186 BondsPLUS product of, 180 –182 StocksPLUS strategy used by, 179–180 bern_z04bindex.qxd 276 4/3/07 8:20 AM Page 276 INDEX Pioneering Investment Management (Swensen), 152, 163–164 Platinum Grove Asset Management CTAs transaction by, 119–120 omega risk transfer approach used by, 111–112, 118 origins of, 110 –111 performance of, 111, 119–123 risk transfer strategies employed by, 118–124 Scholes’ description of, 124 Plexus Group, 54 Policy Portfolio described, 211 Yale University endowment use of, 156–163 Portable alpha asset allocation goes 2x strategy for, 190 –191 Asset Trust platform strategy for, 191–192, 193–194 Damsma’s strategies for, 184 –195 focus on generating excess returns using, 244 growing interest in, 176 Pimco’s strategies for, 182–183 Portfolio insurance strategy, 53 Portfolio management Fundamental Law of Active Management of, 138, 139, 140 Litterman’s approach to, 221–222, 228–232 REITs (real estate investment trusts), 84, 202–205, 211 risk as central to, 218–219, 237 Swensen’s approach to Yale endowment, 153–163 Portfolio performance of Barclays Global Investors ( BGI), 129–130, 140 –141, 142–143, 146 of Goldman Sachs, 216–217, 221–223, 230, 233–234 Markowitz’s theory on selecting/predicting, 100, 103, 105–109 mutual funds, 21–23, 128, 133 of Platinum Grove, 111, 119–123 of REITs (real estate investment trusts), 207–209 relationship between alphas and, 208 simulations to predict, 101–104, 162 of Yale University endowment, 148–150, 154 –158, 160 –161, 162–163 See also Returns Portfolios eliminating benchmarks to create long and short, 230 gains in Taiwan stock market (1990s), 56–57 Leibowitz’s work with CREF, 200 –202 Litterman on combining long and short position of, 230 –232 Litterman on Sharpe Ratios of, 221–222 Litterman on shifting from individual assets to, 228–229 Markowitz’s theory on selecting, 100, 103, 105–109 Merton’s Country A and B scenario using, 55–56 replicating, 52–53, 54 Samuelson on diversif ication of, 42 bern_z04bindex.qxd 4/3/07 8:20 AM Page 277 Index Treynor and Black on optimal selection in, 185 U.S. equities, 160 –161, 202, 203, 204, 205 See also Institutional investing “Portfolio Selection” (Markowitz), 103, 225 Portfolio Selection theory, 100, 103, 105–108 Positive momentum blue noise, 41 Pricing Black-Scholes-Merton options model on, 52, 67, 85, 110, 195 Capital Asset Pricing Model on, 10, 19, 24, 38, 48, 67 importance and signif icance of, 113 irrational, 28–30 Leibowitz’s ability to calculate bond, 199–200 risk change and impact on, 115 Sharpe’s algebraic specif ication of CAPM asset, 167–168 Shiller on excess volatility of market, 68–71 Shiller and Jung’s study on dividend/price ratio, 74 –76 See also Financial markets Principles of Economics (Marshall), 3 Psychohistory theory of human behavior, 59, 60 Put options look-back, 54 portfolio insurance strategy using, 53 Quantitative Financial Strategies, Inc., 32 Quantitative Investment Strategies, 114 277 Quasi-Rational Economics (Thaler), 15 Quebec Separatists (Canada), 121 A Random Walk Down Wall Street (Malkiel), 21 Rational-agent model bounded, 15–16 Capital Ideas supported by, 10 critique of the, 4 –5, 30 empirical failures of, 6 Kahneman on failures of, 71 Rational bubbles, 29–30 Rational expectations model, 66– 68 Real estate risk, 82–84 Regulation T ( Federal Reserve), 105 REITs (real estate investment trusts) described, 84, 211 Leibowitz’s calculation of passive alphas in, 206–209 Leibowitz’s early work on risk of, 202–205 Renaissance Investment Management, 31–32 Replicating portfolios, 52–53, 54 Retirement Economics, 96–99 Returns alpha expected difference in actual/beta, 38, 92 CAPM on expected, 92–93 equilibrium of risk and, 215, 216, 218, 220, 224 –228, 234, 244 Kurz on trade-offs of risk and, 71 Leibowitz’s return/covariance matrix on, 203–204 Leibowitz on trade-off between risk and, 212 liquidity premium, 118 Markowitz’s optimizing trade-off of risk and, 166–167 bern_z04bindex.qxd 278 4/3/07 8:20 AM Page 278 INDEX Returns (Continued) Monte Carlo simulation on, 97, 162 portable alpha focus on generating excess, 244 separating alpha from beta sources of, 173–177 Treynor on systematic errors providing excess, 24 See also Portfolio performance Risk calculating hedge fund, 24 equilibrium of return and, 215, 218 impact on pricing by changing, 115 Kurz on trade-offs of return and, 71 Leibowitz’s list of dragon, 210, 211, 225 Leibowitz on trade-off between return and, 212 Markowitz’s optimizing trade-off of return and, 166–167 momentum, 27 portfolio management and, 218–219, 237 real estate, 82–84 of selecting stocks, 219 Shiller’s instruments to hedge against, 82–86 trade-off between expected return and, 31 See also Alphas ( beating the market); Beta (systematic risk) Risk management Capital Ideas on applications of, 237 diversif ication for, 42, 92–93, 154, 155, 159, 173 Litterman’s approach to, 221–223 Risk transfer forecasting versus, 123–124 hedging bets by, 114 of longer-term and shorter-term bonds, 121–123 miller-farmer example of, 115–117 omega, 111–112, 118 providing liquidity similarity to, 118 Scholes’ perspective of, 117–124 Rosenberg, Barr BARRA started by, 137 on confronting uncertainty, 11–12 multifactor models designed by, 143–144, 171 theory into practice by, 237 Rosengarten, Jacob, 216 Ross, Stephen, 26, 27, 201, 244 Royal Dutch Petroleum, 27 Rubinstein, Mark, 108–109, 237 Salomon Brothers & Hutzler, 76, 150, 151, 196, 198–200 S&P 500 as benchmark, 38, 140, 186 compared to Yale portfolio growth, 148–149 diversif ication of, 107 StocksPLUS exposure in, 182 StocksPLUS successful beating of, 179–180 synthetic bond strategy to outperform the, 188–191 Yale’s tracking error against, 160 Samuelson, Paul on Behavioral Finance, 39 on behavior of investors, 39–40 bern_z04bindex.qxd 4/3/07 8:20 AM Page 279 Index on existence of positive alpha, 40 –42 Merton, Lo, and Shiller as protégés of, 42–43 on micro eff iciency, 73 on no easy pickings, 25, 38 “A Note on Measurement of Utility” by, 39 Shiller’s work with, 39, 42, 43, 65 Sarbanes-Oxley, 243 Savage, Leonard, 109 Schneeweis, Thomas, 175 Scholes, Myron alpha and beta excluded by, 112 Black-Scholes-Merton options pricing model by, 52, 67, 85, 110, 195 on CAPM return discrepancy, 132 inf luence on BGI by, 143 LTCM partnership by, 76, 78 professional development of, 110 risk transfer as perceived by, 117–124 Stagecoach Fund designed by, 128, 133 StocksPLUS strategy inf luenced by, 180 valuation of options study by, 52 See also Platinum Grove Asset Management Schumpeter, Joseph, 40, 246 Schwarz, Eduardo, 175 Sellers f inancial market transactions by, 113–114 rational-agent model on, 4 –5, 6, 10, 15–16, 30, 71 See also Behavioral Finance; Capital Ideas 279 Sharpe, Bill “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk” by, 91 CAPM as perceived by, 92–96 Financial Engines model of, 97–98, 162 on f inancial theory, 144 inf luence on BGI by, 143 Leibowitz’s celebration of, 212–213 Nobel Prize awarded to, 91, 212–213, 239 original case for CAPM by, 91 on portfolio selection, 106 practical applications by, 96–99 Retirement Economics approach by, 96–97 on simplifying Markowitz’s optimizing approach, 166–167 on state-preference theory, 94 –96 on stock valuation and betas, 172 Sharpe-Treynor-Lintner-Mossin CAPM, 165 Shearson-Lehman bond index, 84 Shiller, Robert Capital Ideas applications by, 43, 65– 66, 240 –241 Eff icient Market Hypothesis studies on micro-eff iciency by, 72–76 excess volatility studies by, 68–71, 79–80, 82 f inance as perceived by, 80 –82 inf luence on Swensen by, 164 inf luences on work of, 71–72 instruments to hedge against risk developed by, 82–86 Irrational Exuberance by, 43, 65, 70, 73, 75 bern_z04bindex.qxd 280 4/3/07 8:20 AM Page 280 INDEX Shiller, Robert (Continued) on macro-ineff iciency of markets, 41 Macro Markets: Creating Institutions for Managing Society’s Largest Economic Risks by, 81, 82 The New Financial Order by, 81, 82 professional background of, 65– 66 on rational expectations model, 66– 68 as Samuelson’s protégé, 39, 42, 43, 65 Shleifer, Andrei, 26, 27 Simon, Herbert, 15 Simscript (computer language), 197 Simulations, 101–104, 162 Single-Index Model, origins of, 100 Slow ideas, 25 Small-Cap Value Equity (1996), 16, 17 Small Mid-Cap Core Equity (1996), 16, 17 Small Mid-Cap Growth Equity (1992), 16, 17 Smith, Adam, 246 Smith, Gary, 172 Social Security income, 97 South Sea bubble (1720 –1721), 29 Stagecoach Fund, 128, 132–133 Stanford Research Institute, 197 State-preference theory, 94 –96 Stock markets Chicago Mercantile Exchange, 85 day trading, 56 globalization of, 242–243 NASDAQ, 29–30, 54 New York Stock Exchange, 54, 56 Taiwan, 56–57 understanding drivers of, 146 Stock momentum, 141 Stocks beta and volatility of, 168, 172 dividends of, 74 –76, 79–80 high-quality risk of selecting, 219 short-selling, 143 See also Bubbles StocksPLUS BondsPLUS product of, 180 –182 described, 179, 179–180 f inancing strategies of, 195 optimal selection approach used by, 185 S&P 500 beat by, 179–180 Scholes’ inf luence on strategy used by, 180 synthetic bond strategy used by, 188–191 See also Gross, Bill Structured notes, 121–122 Swensen, David alpha used as part of portfolio strategy of, 169–170 on Behavioral Finance, 164 Capital Ideas applications by, 154 –164 diversif ication strategy used by, 154, 155, 159 f inancial philosophy of, 161 IBM-World Bank f inancial swap transaction by, 151 on market eff iciency, 244 mean/variance approach by, 155–156, 162–163 Modern Portfolio developed by, 154 –163 Monte Carlo simulations used by, 162 on personnel relationships, 161–162 Pioneering Investment Management by, 152, 163–164 bern_z04bindex.qxd 4/3/07 8:20 AM Page 281 Index portfolio strategies used by, 153–163 Salomon Brothers experience by, 150 –151 Shiller’s inf luence on, 164 Yale Investment Committee relationship with, 152–153 as Yale University endowment CIO, 148–150 See also Yale University endowment Synthetic bond strategy, 188–191 Taiwan stock market, 56–57 Tech bubble (1990s), 145–147, 245 Temin, Peter, 29 Thaler, Richard anomalous behaviors listed by, 15 on house money effect, 8 professional background and interests of, 14 –16 rationality ideas of, 15–16 See also Fuller & Thaler Theories of choice Kahneman’s failure of invariance in, 6–8 Kahneman and Tversky’s work on, 4 –5 Theory of Rational Beliefs ( Kurz), 71, 72 TIAA-CREF 50-50 choice used by, 9–10 Leibowitz’s work with, 196, 200 –202 TIPs (inf lation-protected bonds), 12 Tobin, James, 105, 106–107, 151 Transaction cost management, 223–224 Treynor, Jack on asset valuation, 167 on CAPM, 144, 166, 171 281 on challenges of CAPM, 171 on optimal portfolio selection, 185 on reaching equilibrium, 245 recommendations for Yale portfolio management by, 152–153 on search for alpha, 174 –175 on slow ideas, 25 on systematic errors producing excess returns, 24 “The Trouble with Econometric Models” ( Black), 215 Tversky, Amos inf luence on Behavioral Finance by, 10 theory of choice work by, 4 Uncertainty Heisenberg Uncertainty Principle, 244 Rosenberg on confronting, 11–12 Universitat Pompeu Fabra ( Barcelona), 29 University of Chicago, 201 U.S. bonds BondsPLUS, 180 –182, 195 f ixed-income management of, 200 Leibowitz on beta and volatility of, 206 Leibowitz’s return/covariance matrix on, 203–204 Leibowitz’s yield book on prices of, 199–200 risk transfer and, 121–123 U.S. equities Leibowitz’s return/covariance matrix on, 202, 203–204 reductions of institution use of, 205–206 Swensen’s strategies for, 160 –161 bern_z04bindex.qxd 4/3/07 8:20 AM Page 282 282 INDEX U.S. f inancial markets Capital Ideas observations on, 241 globalization of, 242–243 U.S.

Lo studies with an unusual intensity and a hunger to learn. The possibility that the capital markets are not a random walk came to him quite by accident—in fact, it came to him as he was working on the opposite hypothesis that markets are a random walk. When the evidence fell short of supporting the random walk hypothesis, Lo just looked harder in search of an explanation. bern_c05.qxd 62 3/23/07 9:02 AM Page 62 THE THEORETICIANS After six years, he relates, “I finally decided that markets don’t really follow random walks. The notion is a great idealization but not the real thing. And this work got me tenure at MIT!” One of the results of that extended period of study and experimentation was a book aptly titled, A Non-Random Walk Down Wall Street, coauthored with A. Craig MacKinlay and published in 1999. Frustrated with the shortcomings of Behavioral Finance, but also convinced that the theoretical structure of the Efficient Market Hypothesis has profound f laws in terms of the real world, Lo returned to his original fascination with Isaac Asimov and psychohistory.


pages: 571 words: 105,054

Advances in Financial Machine Learning by Marcos Lopez de Prado

algorithmic trading, Amazon Web Services, asset allocation, backtesting, bioinformatics, Brownian motion, business process, Claude Shannon: information theory, cloud computing, complexity theory, correlation coefficient, correlation does not imply causation, diversification, diversified portfolio, en.wikipedia.org, fixed income, Flash crash, G4S, implied volatility, information asymmetry, latency arbitrage, margin call, market fragmentation, market microstructure, martingale, NP-complete, P = NP, p-value, paper trading, pattern recognition, performance metric, profit maximization, quantitative trading / quantitative finance, RAND corporation, random walk, risk-adjusted returns, risk/return, selection bias, Sharpe ratio, short selling, Silicon Valley, smart cities, smart meter, statistical arbitrage, statistical model, stochastic process, survivorship bias, transaction costs, traveling salesman

This is useful in that bid-ask spreads are a function of liquidity, hence Roll's model can be seen as an early attempt to measure the liquidity of a security. Consider a mid-price series {mt}, where prices follow a Random Walk with no drift, hence price changes Δmt = mt − mt − 1 are independently and identically drawn from a Normal distribution These assumptions are, of course, against all empirical observations, which suggest that financial time series have a drift, they are heteroscedastic, exhibit serial dependency, and their returns distribution is non-Normal. But with a proper sampling procedure, as we saw in Chapter 2, these assumptions may not be too unrealistic. The observed prices, {pt}, are the result of sequential trading against the bid-ask spread: where c is half the bid-ask spread, and bt ∈ { − 1, 1} is the aggressor side.

Maddala and Kim [1998], and Breitung [2014] offer good overviews of the literature. 17.4.1 Chow-Type Dickey-Fuller Test A family of explosiveness tests was inspired by the work of Gregory Chow, starting with Chow [1960]. Consider the first order autoregressive process where ϵt is white noise. The null hypothesis is that yt follows a random walk, H0: ρ = 1, and the alternative hypothesis is that yt starts as a random walk but changes at time τ*T, where τ* ∈ (0, 1), into an explosive process: At time T we can test for a switch (from random walk to explosive process) having taken place at time τ*T (break date). In order to test this hypothesis, we fit the following specification, where Dt[τ*] is a dummy variable that takes zero value if t < τ*T, and takes the value one if t ≥ τ*T. Then, the null hypothesis H0: δ = 0 is tested against the (one-sided) alternative H1: δ > 1: The main drawback of this method is that τ* is unknown.

As Breitung [2014] explains, we should leave out some of the possible τ* at the beginning and end of the sample, to ensure that either regime is fitted with enough observations (there must be enough zeros and enough ones in Dt[τ*]). The test statistic for an unknown τ* is the maximum of all T(1 − 2τ0) values of . Another drawback of Chow's approach is that it assumes that there is only one break date τ*T, and that the bubble runs up to the end of the sample (there is no switch back to a random walk). For situations where three or more regimes (random walk → bubble → random walk …) exist, we need to discuss the Supremum Augmented Dickey-Fuler (SADF) test. 17.4.2 Supremum Augmented Dickey-Fuller In the words of Phillips, Wu and Yu [2011], “standard unit root and cointegration tests are inappropriate tools for detecting bubble behavior because they cannot effectively distinguish between a stationary process and a periodically collapsing bubble model.


Stocks for the Long Run, 4th Edition: The Definitive Guide to Financial Market Returns & Long Term Investment Strategies by Jeremy J. Siegel

addicted to oil, asset allocation, backtesting, Black-Scholes formula, Bretton Woods, business cycle, buy and hold, buy low sell high, California gold rush, capital asset pricing model, cognitive dissonance, compound rate of return, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, dividend-yielding stocks, dogs of the Dow, equity premium, Eugene Fama: efficient market hypothesis, Everybody Ought to Be Rich, fixed income, German hyperinflation, implied volatility, index arbitrage, index fund, Isaac Newton, joint-stock company, Long Term Capital Management, loss aversion, market bubble, mental accounting, Myron Scholes, new economy, oil shock, passive investing, Paul Samuelson, popular capitalism, prediction markets, price anchoring, price stability, purchasing power parity, random walk, Richard Thaler, risk tolerance, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, South Sea Bubble, stocks for the long run, survivorship bias, technology bubble, The Great Moderation, The Wisdom of Crowds, transaction costs, tulip mania, Vanguard fund

In 1965, Professor Paul Samuelson of MIT showed that the randomness in security prices did not contradict the laws of supply and demand.4 In fact, such randomness was a result of a free and efficient market in which investors had already incorporated all the known factors influencing the price of the stock. This is the crux of the efficient market hypothesis. If the market is efficient, prices will change only when new, unanticipated information is released to the market. Since unanticipated information is as likely to be good as it is to be bad, the resulting movement in stock prices is random. Price charts will look like a random walk since the probability that stocks go up or down is completely random and cannot be predicted.5 SIMULATIONS OF RANDOM STOCK PRICES If stock prices are indeed random, their movements should not be distinguishable from counterfeits generated randomly by a computer.

TABLE 2–2 Portfolio Allocation: Percentage of Portfolio Recommended in Stocks Based on All Historical Data Risk Tolerance 1 Year Holding Period 5 Years 10 Years 30 Years Ultraconservative (Minimum Risk) 9.0% 22.0% 39.3% 71.4% Conservative 25.0% 38.7% 59.6% 89.5% Moderate 50.0% 61.6% 88.0% 116.2% Aggressive Risk Taker 75.0% 78.5% 110.1% 139.1% CHAPTER 2 Risk, Return, and Portfolio Allocation 35 because modern portfolio theory was established when the academic profession believed in the random walk theory of security prices. As noted earlier, under a random walk, the relative risk of various securities does not change for different holding periods, so portfolio allocations do not depend on how long one holds the asset. The holding period becomes a crucial issue in portfolio theory when the data reveal the mean reversion of stock returns.8 INFLATION-INDEXED BONDS Until the last decade, there was no U.S. government bond whose return was guaranteed against changes in the price level. But in January 1997, the U.S. Treasury issued the first government-guaranteed inflation-indexed bond. The coupons and principal repayment of this inflation-protected bond are automatically increased when the price level rises, so bondholders suffer no loss of purchasing power when they receive the coupons or final principal.

With the remaining seven charts, the brokers showed 4 Paul Samuelson, “Proof That Properly Anticipated Prices Fluctuate Randomly,” Industrial Management Review, vol. 6 (1965), p. 49. 5 More generally, the sum of the product of each possible price change times the probability of its occurrence is zero. This is called a martingale, of which a random walk (50 percent probability up, 50 percent probability down) is a special case. CHAPTER 17 Technical Analysis and Investing with the Trend FIGURE 293 17–1 Real and Simulated Stock Indexes Figure A Figure B Figure C Figure D Figure E Figure F Figure G Figure H 294 PART 4 Stock Fluctuations in the Short Run no ability to distinguish actual from counterfeit data. The true historical prices are represented by charts b, d, e, and h, while the computer-generated data are charts a, c, f, and g.6 TRENDING MARKETS AND PRICE REVERSALS Despite the fact that many “trends” are in fact the result of the totally random movement of stock prices, many traders will not invest against a trend that they believe they have identified.


Monte Carlo Simulation and Finance by Don L. McLeish

Black-Scholes formula, Brownian motion, capital asset pricing model, compound rate of return, discrete time, distributed generation, finite state, frictionless, frictionless market, implied volatility, incomplete markets, invention of the printing press, martingale, p-value, random walk, Sharpe ratio, short selling, stochastic process, stochastic volatility, survivorship bias, the market place, transaction costs, value at risk, Wiener process, zero-coupon bond, zero-sum game

Equivalently, we generate X = V1 /V2 where Vi ∼ U [−1, 1] conditional on V12 + V22 · 1 to produce a standard Cauchy variate X. Example: Stable random walk. A stable random walk may be used to model a stock price but the closest analogy to the Black Scholes model would be a logstable process St under which the distribution of ln(St ) has a symmetric stable distribution. Unfortunately, this specification renders impotent many of our tools of analysis, since except in 158 CHAPTER 3. BASIC MONTE CARLO METHODS the case α = 2 or the case β = −1, such a stock price process St has no finite moments at all. Nevertheless, we may attempt to fit stable laws to the distribution of ln(St ) for a variety of stocks and except in the extreme tails, symmetric stable laws with index α ' 1.7 often provide a reasonably good fit. To see what such a returns process looks like, we generate a random walk with 10,000 time steps where each increment is distributed as independent stable random variables having parameter 1.7.

It has been an important part of the literature in Physics, Probability and Finance at least since the papers of Bachelier and Einstein, about 100 years ago. A Brownian motion process also has some interesting and remarkable theoretical properties; it is continuous with probability one but the probability that the process has finite 10 68 CHAPTER 2. SOME BASIC THEORY OF FINANCE Random Walk 4 3 2 Sn 1 0 -1 -2 -3 0 2 4 6 8 10 n 12 14 16 18 Figure 2.7: A sample path of a Random Walk variation in any interval is 0. With probability one it is nowhere differentiable. Of course one might ask how a process with such apparently bizarre properties can be used to approximate real-world phenomena, where we expect functions to be built either from continuous and differentiable segments or jumps in the process. The answer is that a very wide class of functions constructed from those that are quite well-behaved (e.g. step functions) and that have independent increments converge as the scale on which they move is refined either to a Brownian motion process or to a process defined as an integral with respect to a Brownian motion process and so this is a useful approximation to a broad range of continuous time processes.

The answer is that a very wide class of functions constructed from those that are quite well-behaved (e.g. step functions) and that have independent increments converge as the scale on which they move is refined either to a Brownian motion process or to a process defined as an integral with respect to a Brownian motion process and so this is a useful approximation to a broad range of continuous time processes. For example, consider a random walk process Pn Sn = i=1 Xi where the random variables Xi are independent identically distributed with expected value E(Xi ) = 0 and var(Xi ) = 1. Suppose we plot the graph of this random walk (n, Sn ) as below. Notice that we have linearly interpolated the graph so that the function is defined for all n, whether integer or not. [FIGURE 2.7 ABOUT HERE] 20 MODELS IN CONTINUOUS TIME 69 Now if we increase the sample size and decrease the scale appropriately on both axes, the result is, in the limit, a Brownian motion process.


pages: 1,829 words: 135,521

Python for Data Analysis: Data Wrangling with Pandas, NumPy, and IPython by Wes McKinney

business process, Debian, Firefox, general-purpose programming language, Google Chrome, Guido van Rossum, index card, p-value, quantitative trading / quantitative finance, random walk, recommendation engine, sentiment analysis, side project, sorting algorithm, statistical model, type inference

Partial list of numpy.random functionsFunctionDescription seed Seed the random number generator permutation Return a random permutation of a sequence, or return a permuted range shuffle Randomly permute a sequence in-place rand Draw samples from a uniform distribution randint Draw random integers from a given low-to-high range randn Draw samples from a normal distribution with mean 0 and standard deviation 1 (MATLAB-like interface) binomial Draw samples from a binomial distribution normal Draw samples from a normal (Gaussian) distribution beta Draw samples from a beta distribution chisquare Draw samples from a chi-square distribution gamma Draw samples from a gamma distribution uniform Draw samples from a uniform [0, 1) distribution 4.7 Example: Random Walks The simulation of random walks provides an illustrative application of utilizing array operations. Let’s first consider a simple random walk starting at 0 with steps of 1 and –1 occurring with equal probability. Here is a pure Python way to implement a single random walk with 1,000 steps using the built-in random module: In [247]: import random .....: position = 0 .....: walk = [position] .....: steps = 1000 .....: for i in range(steps): .....: step = 1 if random.randint(0, 1) else -1 .....: position += step .....: walk.append(position) .....: See Figure 4-4 for an example plot of the first 100 values on one of these random walks: In [249]: plt.plot(walk[:100]) Figure 4-4. A simple random walk You might make the observation that walk is simply the cumulative sum of the random steps and could be evaluated as an array expression.

Thus, I use the np.random module to draw 1,000 coin flips at once, set these to 1 and –1, and compute the cumulative sum: In [251]: nsteps = 1000 In [252]: draws = np.random.randint(0, 2, size=nsteps) In [253]: steps = np.where(draws > 0, 1, -1) In [254]: walk = steps.cumsum() From this we can begin to extract statistics like the minimum and maximum value along the walk’s trajectory: In [255]: walk.min() Out[255]: -3 In [256]: walk.max() Out[256]: 31 A more complicated statistic is the first crossing time, the step at which the random walk reaches a particular value. Here we might want to know how long it took the random walk to get at least 10 steps away from the origin 0 in either direction. np.abs(walk) >= 10 gives us a boolean array indicating where the walk has reached or exceeded 10, but we want the index of the first 10 or –10. Turns out, we can compute this using argmax, which returns the first index of the maximum value in the boolean array (True is the maximum value): In [257]: (np.abs(walk) >= 10).argmax() Out[257]: 37 Note that using argmax here is not always efficient because it always makes a full scan of the array. In this special case, once a True is observed we know it to be the maximum value. Simulating Many Random Walks at Once If your goal was to simulate many random walks, say 5,000 of them, you can generate all of the random walks with minor modifications to the preceding code.

, Introspection-Introspection %quickref magic function, About Magic Commands quicksort method, Alternative Sort Algorithms quotation marks in strings, Strings R r character prefacing quotes, Strings R language, pandas, statsmodels, Handling Missing Data radd method, Arithmetic methods with fill values rand function, Pseudorandom Number Generation randint function, Pseudorandom Number Generation randn function, Boolean Indexing, Pseudorandom Number Generation random module, Pseudorandom Number Generation-Simulating Many Random Walks at Once random number generation, Pseudorandom Number Generation-Pseudorandom Number Generation random sampling and permutation, Example: Random Sampling and Permutation random walks example, Example: Random Walks-Simulating Many Random Walks at Once RandomState class, Pseudorandom Number Generation range function, range, Creating ndarrays rank method, Sorting and Ranking ranking data in pandas library, Sorting and Ranking-Sorting and Ranking ravel method, Reshaping Arrays rc method, matplotlib Configuration rdiv method, Arithmetic methods with fill values re module, Functions Are Objects, Regular Expressions read method, Files and the Operating System-Files and the Operating System read-and-write mode for files, Files and the Operating System read-only mode for files, Files and the Operating System reading datain Microsoft Excel files, Reading Microsoft Excel Files-Reading Microsoft Excel Files in text format, Reading and Writing Data in Text Format-Reading Text Files in Pieces readline functionality, Searching and Reusing the Command History readlines method, Files and the Operating System read_clipboard function, Reading and Writing Data in Text Format read_csv function, Files and the Operating System, Reading and Writing Data in Text Format, Reading and Writing Data in Text Format, Bar Plots, Column-Wise and Multiple Function Application read_excel function, Reading and Writing Data in Text Format, Reading Microsoft Excel Files read_feather function, Reading and Writing Data in Text Format read_fwf function, Reading and Writing Data in Text Format read_hdf function, Reading and Writing Data in Text Format, Using HDF5 Format read_html function, Reading and Writing Data in Text Format, XML and HTML: Web Scraping-Parsing XML with lxml.objectify read_json function, Reading and Writing Data in Text Format, JSON Data read_msgpack function, Reading and Writing Data in Text Format read_pickle function, Reading and Writing Data in Text Format, Binary Data Formats read_sas function, Reading and Writing Data in Text Format read_sql function, Reading and Writing Data in Text Format, Interacting with Databases read_stata function, Reading and Writing Data in Text Format read_table function, Reading and Writing Data in Text Format, Reading and Writing Data in Text Format, Working with Delimited Formats reduce method, ufunc Instance Methods reduceat method, ufunc Instance Methods reductions (aggregations), Mathematical and Statistical Methods references in Python, Variables and argument passing-Dynamic references, strong types regplot method, Scatter or Point Plots regress function, Example: Group-Wise Linear Regression regular expressionspasses as delimiters, Reading and Writing Data in Text Format string manipulation and, Regular Expressions-Regular Expressions reindex method, Reindexing-Reindexing, Selection with loc and iloc, Axis Indexes with Duplicate Labels, Upsampling and Interpolation reload function, Reloading Module Dependencies remove method, Adding and removing elements, set remove_categories method, Categorical Methods remove_unused_categories method, Categorical Methods rename method, Renaming Axis Indexes rename_categories method, Categorical Methods reorder_categories method, Categorical Methods repeat function, Repeating Elements: tile and repeat repeat method, Vectorized String Functions in pandas replace method, Replacing Values, String Object Methods-String Object Methods, Vectorized String Functions in pandas requests package, Interacting with Web APIs resample method, Date Ranges, Frequencies, and Shifting, Resampling and Frequency Conversion-Open-High-Low-Close (OHLC) resampling, Grouped Time Resampling resamplingdefined, Resampling and Frequency Conversion downsampling and, Resampling and Frequency Conversion-Open-High-Low-Close (OHLC) resampling OHLC, Open-High-Low-Close (OHLC) resampling upsampling and, Resampling and Frequency Conversion, Upsampling and Interpolation with periods, Resampling with Periods %reset magic function, About Magic Commands, Input and Output Variables reset_index method, Pivoting “Wide” to “Long” Format, Returning Aggregated Data Without Row Indexes reshape method, Fancy Indexing, Reshaping Arrays *rest syntax, Unpacking tuples return statement, Functions reusing command history, Searching and Reusing the Command History reversed function, reversed rfind method, String Object Methods rfloordiv method, Arithmetic methods with fill values right join type, Database-Style DataFrame Joins rint function, Universal Functions: Fast Element-Wise Array Functions rjust method, String Object Methods rmul method, Arithmetic methods with fill values rollback method, Shifting dates with offsets rollforward method, Shifting dates with offsets rolling function, Moving Window Functions, Moving Window Functions rolling_corr function, Binary Moving Window Functions row major order (C order), C Versus Fortran Order, The Importance of Contiguous Memory row_stack function, Concatenating and Splitting Arrays rpow method, Arithmetic methods with fill values rstrip method, String Object Methods, Vectorized String Functions in pandas rsub method, Arithmetic methods with fill values %run magic functionabout, About Magic Commands exceptions and, Exceptions in IPython interactive debugger and, Interactive Debugger, Other ways to make use of the debugger IPython and, The Python Interpreter, The %run Command-Interrupting running code reusing command history with, Searching and Reusing the Command History r_ object, Stacking helpers: r_ and c_ S %S datetime format, Dates and times, Converting Between String and Datetime s(tep) debugger command, Interactive Debugger sample method, Permutation and Random Sampling, Example: Random Sampling and Permutation save function, File Input and Output with Arrays, Advanced Array Input and Output savefig method, Saving Plots to File savez function, File Input and Output with Arrays savez_compressed function, File Input and Output with Arrays scalar types in Python, Scalar Types-Dates and times, Arithmetic with NumPy Arrays scatter plot matrix, Scatter or Point Plots scatter plots, Scatter or Point Plots scikit-learn library, scikit-learn, Introduction to scikit-learn-Introduction to scikit-learn SciPy library, SciPy scope of functions, Namespaces, Scope, and Local Functions scripting languages, Why Python for Data Analysis?


pages: 389 words: 98,487

The Undercover Economist: Exposing Why the Rich Are Rich, the Poor Are Poor, and Why You Can Never Buy a Decent Used Car by Tim Harford

Albert Einstein, barriers to entry, Berlin Wall, business cycle, collective bargaining, congestion charging, Corn Laws, David Ricardo: comparative advantage, decarbonisation, Deng Xiaoping, Fall of the Berlin Wall, George Akerlof, information asymmetry, invention of movable type, John Nash: game theory, John von Neumann, Kenneth Arrow, Kickstarter, market design, Martin Wolf, moral hazard, new economy, Pearl River Delta, price discrimination, Productivity paradox, race to the bottom, random walk, rent-seeking, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Reagan, sealed-bid auction, second-price auction, second-price sealed-bid, Shenzhen was a fishing village, special economic zone, spectrum auction, The Market for Lemons, Thomas Malthus, trade liberalization, Vickrey auction

In fact, rational investors should be able to second-guess any predictable movements in the stock market or in the price of any particular share—if it’s predictable then, given the money at stake, they will predict it. But that means that if investors really are rational, there won’t be any predictable share movements at all. All the predictability should be sucked out of the stock market very quickly because all trends will be anticipated. The only thing that is left is unpredictable news. As a result of the fact that only random news moves share prices, those prices, and the indices measuring the stock market as a whole, should fluctuate completely at random. Math- • 138 • R A T I O N A L I N S A N I T Y ematicians call the behavior “a random walk”—equally likely on any day to rise as to fall. More correctly, the stock market should exhibit a “random walk with a trend,” meaning that it should on average edge up as the months go past, so that it is competitive compared with other potential investments such as money in a savings account, or property.

But if enough experts knew that, it wouldn’t be the shortest any more. The truth is that busy, smart, agile, and experienced shoppers are a bit better at calling the fastest lines and can probably average a quicker time than the rest of us. But not by much. Value and price— beyond the random walk Assuming that what is true of supermarket lines is also true of stock-market prices, economists should be able to throw some light on the market, but not very much. Many economists do work for investment funds. They are as wrong nearly as often as they are right, but not quite. Our modified random walk theory tells us that this is what we should expect. So, what do these economists do to provide investment funds with such tiny edges over the market? The starting point is to view stock shares for what they are: a claim on the future profits of a company.

The answer would be useful, not least because Amazon’s roller-coaster performance is common. So can the Undercover Economist say anything about why share prices acted the way they did, and how they might behave in the future? A random walk Economists face a serious problem in trying to say anything sensible about stock prices. Economists work by studying rational behavior, but the more rational the behavior of stock-market investors, the more erratic the behavior of the stock market becomes. Here’s why. Rational people would buy shares today if it was obvious that they would go up tomorrow, and sell them if it was obvious that they would fall. But this means that any forecast that shares will obviously rise tomorrow will be wrong: shares will rise today instead because people will buy them, and keep buying them until they are no longer so cheap that they will obviously rise tomorrow.


pages: 1,535 words: 337,071

Networks, Crowds, and Markets: Reasoning About a Highly Connected World by David Easley, Jon Kleinberg

Albert Einstein, AltaVista, clean water, conceptual framework, Daniel Kahneman / Amos Tversky, Douglas Hofstadter, Erdős number, experimental subject, first-price auction, fudge factor, George Akerlof, Gerard Salton, Gerard Salton, Gödel, Escher, Bach, incomplete markets, information asymmetry, information retrieval, John Nash: game theory, Kenneth Arrow, longitudinal study, market clearing, market microstructure, moral hazard, Nash equilibrium, Network effects, Pareto efficiency, Paul Erdős, planetary scale, prediction markets, price anchoring, price mechanism, prisoner's dilemma, random walk, recommendation engine, Richard Thaler, Ronald Coase, sealed-bid auction, search engine result page, second-price auction, second-price sealed-bid, Simon Singh, slashdot, social web, Steve Jobs, stochastic process, Ted Nelson, The Market for Lemons, The Wisdom of Crowds, trade route, transaction costs, ultimatum game, Vannevar Bush, Vickrey auction, Vilfredo Pareto, Yogi Berra, zero-sum game

LINK ANALYSIS AND WEB SEARCH Given that the two formulations of PageRank — based on repeated improvement and random walks respectively — are equivalent, we do not strictly speaking gain anything at a formal level by having this new definition. But the analysis in terms of random walks provides some additional intuition for PageRank as a measure of importance: the PageRank of a page X is the limiting probability that a random walk across hyperlinks will end up at X, as we run the walk for larger and larger numbers of steps. This equivalent definition using random walks also provides a new and sometimes useful perspective for thinking about some of the issues that came up earlier in the section. For example, the “leakage” of PageRank to nodes F and G in Figure 14.8 has a natural interpretation in terms of the random walk on the network: in the limit, as the walk runs for more and more steps, the probability of the walk reaching F or G is converging to 1; and once it reaches either F or G, it is stuck at these two nodes forever.

. . . . . . . . . . . . . . . . 299 10.6 Advanced Material: A Proof of the Matching Theorem . . . . . . . . . . . . 300 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11 Network Models of Markets with Intermediaries 319 11.1 Price-Setting in Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.2 A Model of Trade on Networks . . . . . . . . . . . . . . . . . . . . . . . . . 323 11.3 Equilibria in Trading Networks . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.4 Further Equilibrium Phenomena: Auctions and Ripple Effects . . . . . . . . 334 11.5 Social Welfare in Trading Networks . . . . . . . . . . . . . . . . . . . . . . . 338 11.6 Trader Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 11.7 Reflections on Trade with Intermediaries . . . . . . . . . . . . . . . . . . . . 342 11.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12 Bargaining and Power in Networks 347 12.1 Power in Social Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.2 Experimental Studies of Power and Exchange . . . . . . . . . . . . . . . . . 350 12.3 Results of Network Exchange Experiments . . . . . . . . . . . . . . . . . . . 352 12.4 A Connection to Buyer-Seller Networks . . . . . . . . . . . . . . . . . . . . . 356 12.5 Modeling Two-Person Interaction: The Nash Bargaining Solution . . . . . . 357 12.6 Modeling Two-Person Interaction: The Ultimatum Game . . . . . . . . . . . 360 12.7 Modeling Network Exchange: Stable Outcomes . . . . . . . . . . . . . . . . 362 12.8 Modeling Network Exchange: Balanced Outcomes . . . . . . . . . . . . . . . 366 12.9 Advanced Material: A Game-Theoretic Approach to Bargaining . . . . . . . 369 12.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 IV Information Networks and the World Wide Web 381 13 The Structure of the Web 383 13.1 The World Wide Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 13.2 Information Networks, Hypertext, and Associative Memory . . . . . . . . . . 386 13.3 The Web as a Directed Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 394 13.4 The Bow-Tie Structure of the Web . . . . . . . . . . . . . . . . . . . . . . . 397 13.5 The Emergence of Web 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 6 CONTENTS 14 Link Analysis and Web Search 405 14.1 Searching the Web: The Problem of Ranking . . . . . . . . . . . . . . . . . . 405 14.2 Link Analysis using Hubs and Authorities . . . . . . . . . . . . . . . . . . . 407 14.3 PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 14.4 Applying Link Analysis in Modern Web Search . . . . . . . . . . . . . . . . 420 14.5 Applications beyond the Web . . . . . . . . . . . . . . . . . . . . . . . . . . 423 14.6 Advanced Material: Spectral Analysis, Random Walks, and Web Search . . . 425 14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 15 Sponsored Search Markets 445 15.1 Advertising Tied to Search Behavior . . . . . . . . . . . . . . . . . . . . . . 445 15.2 Advertising as a Matching Market . . . . . . . . . . . . . . . . . . . . . . . . 448 15.3 Encouraging Truthful Bidding in Matching Markets: The VCG Principle . . 452 15.4 Analyzing the VCG Procedure: Truth-Telling as a Dominant Strategy . . . . 457 15.5 The Generalized Second Price Auction . . . . . . . . . . . . . . . . . . . . . 460 15.6 Equilibria of the Generalized Second Price Auction . . . . . . . . . . . . . . 464 15.7 Ad Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 15.8 Complex Queries and Interactions Among Keywords . . . . . . . . . . . . . 469 15.9 Advanced Material: VCG Prices and the Market-Clearing Property . . . . . 470 15.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 V Network Dynamics: Population Models 489 16 Information Cascades 491 16.1 Following the Crowd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 16.2 A Simple Herding Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 493 16.3 Bayes’ Rule: A Model of Decision-Making Under Uncertainty . . . . . . . . . 497 16.4 Bayes’ Rule in the Herding Experiment . . . . . . . . . . . . . . . . . . . . . 502 16.5 A Simple, General Cascade Model . . . . . . . . . . . . . . . . . . . . . . . . 504 16.6 Sequential Decision-Making and Cascades . . . . . . . . . . . . . . . . . . . 508 16.7 Lessons from Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 17 Network Effects 517 17.1 The Economy Without Network Effects . . . . . . . . . . . . . . . . . . . . . 518 17.2 The Economy with Network Effects . . . . . . . . . . . . . . . . . . . . . . . 522 17.3 Stability, Instability, and Tipping Points . . . . . . . . . . . . . . . . . . . . 525 17.4 A Dynamic View of the Market . . . . . . . . . . . . . . . . . . . . . . . . . 527 17.5 Industries with Network Goods . . . . . . . . . . . . . . . . . . . . . . . . . 534 17.6 Mixing Individual Effects with Population-Level Effects . . . . . . . . . . . . 536 17.7 Advanced Material: Negative Externalities and The El Farol Bar Problem . 541 17.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 CONTENTS 7 18 Power Laws and Rich-Get-Richer Phenomena 553 18.1 Popularity as a Network Phenomenon . . . . . . . . . . . . . . . . . . . . . . 553 18.2 Power Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 18.3 Rich-Get-Richer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 18.4 The Unpredictability of Rich-Get-Richer Effects . . . . . . . . . . . . . . . . 559 18.5 The Long Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 18.6 The Effect of Search Tools and Recommendation Systems . . . . . . . . . . . 564 18.7 Advanced Material: Analysis of Rich-Get-Richer Processes . . . . . . . . . . 565 18.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 VI Network Dynamics: Structural Models 571 19 Cascading Behavior in Networks 573 19.1 Diffusion in Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 19.2 Modeling Diffusion through a Network . . . . . . . . . . . . . . . . . . . . . 575 19.3 Cascades and Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 19.4 Diffusion, Thresholds, and the Role of Weak Ties . . . . . . . . . . . . . . . 588 19.5 Extensions of the Basic Cascade Model . . . . . . . . . . . . . . . . . . . . . 590 19.6 Knowledge, Thresholds, and Collective Action . . . . . . . . . . . . . . . . . 593 19.7 Advanced Material: The Cascade Capacity . . . . . . . . . . . . . . . . . . . 597 19.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 20 The Small-World Phenomenon 621 20.1 Six Degrees of Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 20.2 Structure and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 20.3 Decentralized Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 20.4 Modeling the Process of Decentralized Search . . . . . . . . . . . . . . . . . 629 20.5 Empirical Analysis and Generalized Models . . . . . . . . . . . . . . . . . . 632 20.6 Core-Periphery Structures and Difficulties in Decentralized Search . . . . . . 638 20.7 Advanced Material: Analysis of Decentralized Search . . . . . . . . . . . . . 640 20.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 21 Epidemics 655 21.1 Diseases and the Networks that Transmit Them . . . . . . . . . . . . . . . . 655 21.2 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 21.3 The SIR Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 21.4 The SIS Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 21.5 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 21.6 Transient Contacts and the Dangers of Concurrency . . . . . . . . . . . . . . 672 21.7 Genealogy, Genetic Inheritance, and Mitochondrial Eve . . . . . . . . . . . . 676 21.8 Advanced Material: Analysis of Branching and Coalescent Processes . . . . . 682 21.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 8 CONTENTS VII Institutions and Aggregate Behavior 699 22 Markets and Information 701 22.1 Markets with Exogenous Events . . . . . . . . . . . . . . . . . . . . . . . . . 702 22.2 Horse Races, Betting, and Beliefs . . . . . . . . . . . . . . . . . . . . . . . . 704 22.3 Aggregate Beliefs and the “Wisdom of Crowds” . . . . . . . . . . . . . . . . 710 22.4 Prediction Markets and Stock Markets . . . . . . . . . . . . . . . . . . . . . 714 22.5 Markets with Endogenous Events . . . . . . . . . . . . . . . . . . . . . . . . 717 22.6 The Market for Lemons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 22.7 Asymmetric Information in Other Markets . . . . . . . . . . . . . . . . . . . 724 22.8 Signaling Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 22.9 Quality Uncertainty On-Line: Reputation Systems and Other Mechanisms . 729 22.10Advanced Material: Wealth Dynamics in Markets . . . . . . . . . . . . . . . 732 22.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 23 Voting 745 23.1 Voting for Group Decision-Making . . . . . . . . . . . . . . . . . . . . . . . 745 23.2 Individual Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 23.3 Voting Systems: Majority Rule . . . . . . . . . . . . . . . . . . . . . . . . . 750 23.4 Voting Systems: Positional Voting . . . . . . . . . . . . . . . . . . . . . . . . 755 23.5 Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 758 23.6 Single-Peaked Preferences and the Median Voter Theorem . . . . . . . . . . 760 23.7 Voting as a Form of Information Aggregation . . . . . . . . . . . . . . . . . . 766 23.8 Insincere Voting for Information Aggregation . . . . . . . . . . . . . . . . . . 768 23.9 Jury Decisions and the Unanimity Rule . . . . . . . . . . . . . . . . . . . . . 771 23.10Sequential Voting and the Relation to Information Cascades . . . . . . . . . 776 23.11Advanced Material: A Proof of Arrow’s Impossibility Theorem . . . . . . . . 777 23.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 24 Property Rights 785 24.1 Externalities and the Coase Theorem . . . . . . . . . . . . . . . . . . . . . . 785 24.2 The Tragedy of the Commons . . . . . . . . . . . . . . . . . . . . . . . . . . 790 24.3 Intellectual Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 24.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 Chapter 1 Overview Over the past decade there has been a growing public fascination with the complex “connectedness” of modern society.

Since both PageRank values and random-walk probabilities start out the same (they are initially 1/n for all nodes), and they then evolve according to exactly the same rule, they remain the same forever. This justifies the claim that we made in Section 14.3: Claim: The probability of being at a page X after k steps of this random walk is precisely the PageRank of X after k applications of the Basic PageRank Update Rule. And this makes intuitive sense. Like PageRank, the probability of being at a given node in a random walk is something that gets divided up evenly over all the outgoing links from a given node, and then passed on to the nodes at the other ends of these links. In other words, probability and PageRank both flow through the graph according to the same process. A Scaled Version of the Random Walk. We can also formulate an interpretation of the Scaled PageRank Update Rule in terms of random walks.


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Millionaire Teacher: The Nine Rules of Wealth You Should Have Learned in School by Andrew Hallam

Albert Einstein, asset allocation, Bernie Madoff, buy and hold, diversified portfolio, financial independence, George Gilder, index fund, Long Term Capital Management, new economy, passive investing, Paul Samuelson, Ponzi scheme, pre–internet, price stability, random walk, risk tolerance, Silicon Valley, South China Sea, stocks for the long run, survivorship bias, transaction costs, Vanguard fund, yield curve

When they don’t, it spells trouble. Even shares of the world’s largest technology companies sold at nosebleed prices as they defied business profit levels. And, as shown in Table 4.4, when cold, hard business earnings eventually yanked the price leashes back to Earth, people who had ignored the age-old premise (that business growth and stock growth is directly proportional) eventually lost their shirts. Investing $10,000 in a few of the new millennium’s most popular stocks during 2000 would have resulted in some devastating losses for investors. Table 4.4 How Investors were Punished Source: Morningstar and Burton Malkiel, A Random Walk Down Wall Street, 200312 Formerly Hot Stocks $10,000 Invested at the Market High in 2000 Value of the Same $10,000 at the Low of 2001–2002 Amazon.com $10,000 $700 Cisco Systems $10,000 $990 Corning Inc. $10,000 $100 JDS Uniphase $10,000 $50 Lucent Technologies $10,000 $70 Nortel Networks $10,000 $30 Priceline.com $10,000 $60 Yahoo!

Back in Chapter 4, I showed you a chart of technology companies and how far their share prices fell from 2000 to 2002. In 2000, whose investment report recommended purchasing Nortel Networks <www.nortel.com>, Lucent Technologies <www.alcatel-lucent.com/wps/portal?COUNTRY_CODE=US&COOKIE_SET=false>, JDS Uniphase <www.jdsu.com/en-us/Pages/Home.aspx> and Cisco Systems <www.cisco.com/>? You guessed it: George Gilder’s. Table 8.1 puts the reality in perspective. If you had a total of $40,000 invested in the above four “Gilder-touted” businesses in 2000, it would have dropped to $1,140 by 2002. Table 8.1 Prices of Technology Stocks Plummet (2000–2002) Source: Morningstar and Burton Malkiel’s A Random Walk Guide to Investing High Value in 2000 Low Value in 2002 <Amazon.com> $10,000 $700 Cisco Systems $10,000 $990 Corning Inc. $10,000 $100 JDS Uniphase $10,000 $50 Lucent Technologies $10,000 $70 Nortel Networks $10,000 $30 <Priceline.com> $10,000 $60 Yahoo!

One hundred dollars invested in a broad-based U.S. stock market index from 1994 to 2004 would have turned into roughly $271 after taxes, at 10.5 percent annually. Interestingly, more than 98 percent of invested mutual fund money gets pushed into Morningstar’s top-rated funds25 But choosing which actively managed mutual fund will perform well in the future is, in Burton Malkiel’s words: “. . . like an obstacle course through hell’s kitchen.”26 Malkiel, a professor of economics at Princeton University and the bestselling author of A Random Walk Guide to Investing, adds: There is no way to choose the best [actively managed mutual fund] managers in advance. I have calculated the results of employing strategies of buying the funds with the best recent-year performance, best recent two-year performance, best five-year and ten-year performance, and not one of these strategies produced above average returns. I calculated the returns from buying the best funds selected by Forbes magazine . . . and found that these funds subsequently produced below average returns.27 Still, most financial advisers won’t give up.


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High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

algorithmic trading, asset allocation, asset-backed security, automated trading system, backtesting, Black Swan, Brownian motion, business cycle, business process, buy and hold, capital asset pricing model, centralized clearinghouse, collapse of Lehman Brothers, collateralized debt obligation, collective bargaining, computerized trading, diversification, equity premium, fault tolerance, financial intermediation, fixed income, high net worth, implied volatility, index arbitrage, information asymmetry, interest rate swap, inventory management, law of one price, Long Term Capital Management, Louis Bachelier, margin call, market friction, market microstructure, martingale, Myron Scholes, New Journalism, p-value, paper trading, performance metric, profit motive, purchasing power parity, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, short selling, Small Order Execution System, statistical arbitrage, statistical model, stochastic process, stochastic volatility, systematic trading, trade route, transaction costs, value at risk, yield curve, zero-sum game

As Table 7.4 shows, the runs test rejects randomness of price changes at 1-minute frequencies, except for prices on S&P 500 Depository Receipts (SPY). The results imply strong market inefficiency in 1-minute data for the securities shown. Market inefficiency measured by runs test decreases or disappears entirely at a frequency lower than 10 minutes. Tests of Random Walks Other, more advanced tests for market efficiency have been developed over the years. These tests help traders evaluate the state of the markets and reallocate trading capital to the markets with the most inefficiencies—that is, the most opportunities for reaping profits. When price changes are random, they are said to follow a “random walk.” Formally, a random walk process is specified as follows: ln Pt = ln Pt−1 + εt (7.2) where ln Pt is the logarithm of the price of the financial security of interest at time t, ln Pt-1 is the logarithm of the price of the security one time TABLE 7.4 Non-Parametric Runs Test Applied to Data on Various Securities and Frequencies Recorded on June 8, 2009.

From equation (7.2), log price changes ln Pt are obtained as follows: ln Pt = ln Pt − ln Pt−1 = εt At any given time, the change in log price is equally likely to be positive and negative. The logarithmic price specification ensures that the model does not allow prices to become negative (logarithm of a negative number does not exist). The random walk process can drift, and be specified as shown in equation (7.3): ln Pt = µ + ln Pt−1 + εt (7.3) In this case, the average change in prices equals the drift rather than 0, since ln Pt = ln Pt − ln Pt−1 = µ + εt . The drift can be due to a variety of factors; persistent inflation, for example, would uniformly lower the value of the U.S. dollar, inflicting a small positive drift on prices of all U.S. equities. At very high frequencies, however, drifts are seldom noticeable. Lo and MacKinlay (1988) developed a popular test for whether or not a given price follows a random walk.

Harris, 1988. “Estimating the Components of the Bid-Ask Spread.” Journal of Financial Economics 21, 123–142. Glosten, Lawrence and P. Milgrom, 1985. “Bid, Ask, and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders.” Journal of Financial Economics 13, 71–100. Goettler, R., C. Parlour and U. Rajan, 2005. “Equilibrium in a Dynamic Limit Order Market.” Journal of Finance 60, 2149–2192. Goettler, R., C. Parlour and U. Rajan, 2007. “Microstructure Effects and Asset Pricing.” Working paper, University of California—Berkeley. References 313 Goodhart, Charles A.E., 1988. “The Foreign Exchange Market: A Random Walk with a Dragging Anchor.” Economica 55, 437–460. Goodhart, Charles A.E. and Maureen O’Hara, 1997. “High Frequency Data in Financial Markets: Issues and Applications.” Journal of Empirical Finance 4, 73– 114.


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Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann

Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, banking crisis, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bretton Woods, Brownian motion, business cycle, capital asset pricing model, Cass Sunstein, collective bargaining, colonial exploitation, compound rate of return, conceptual framework, corporate governance, Credit Default Swap, David Ricardo: comparative advantage, debt deflation, delayed gratification, Detroit bankruptcy, disintermediation, diversified portfolio, double entry bookkeeping, Edmond Halley, en.wikipedia.org, equity premium, financial independence, financial innovation, financial intermediation, fixed income, frictionless, frictionless market, full employment, high net worth, income inequality, index fund, invention of the steam engine, invention of writing, invisible hand, James Watt: steam engine, joint-stock company, joint-stock limited liability company, laissez-faire capitalism, Louis Bachelier, mandelbrot fractal, market bubble, means of production, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, new economy, passive investing, Paul Lévy, Ponzi scheme, price stability, principal–agent problem, profit maximization, profit motive, quantitative trading / quantitative finance, random walk, Richard Thaler, Robert Shiller, Robert Shiller, shareholder value, short selling, South Sea Bubble, sovereign wealth fund, spice trade, stochastic process, the scientific method, The Wealth of Nations by Adam Smith, Thomas Malthus, time value of money, too big to fail, trade liberalization, trade route, transatlantic slave trade, tulip mania, wage slave

These solutions to the option pricing problem linked finance and physics together forever afterward. In fact, it turned out that the Black-Scholes option pricing model was the same as a problem in thermodynamics—a “heat” equation, in which molecules—not stock prices—were drifting randomly. The foundation of the science of thermodynamics is entropy—the tendency toward disorder. Time only goes in one direction, and with it, the universe tends toward less organization, not more. The option pricing model is based on the principle of forecasting the range of future outcomes of the stock price by assuming it will follow a random walk that conforms to Regnault’s square-root of time insight. However, the Black-Scholes formula gives a solution to the option price today by mathematically rolling time backward. It reverses entropy. In this, it echoes the most basic trait of finance—it uses mathematics to transcend time.

For example, we saw in Chapter 15 that Jules Regnault proposed and tested a stochastic process that varied randomly, which resulted in a rule about risk increasing with the square root of time. Likewise, Louis Bachelier more formally developed a random-walk stochastic process. Paul Lévy formalized these prior random walk models into a very general family of stochastic processes referred to as Lévy processes. Brownian motion was just one process in the family of Lévy processes—and perhaps the best behaved of them. Other stochastic processes have such things as discontinuous jumps and unusually large shocks (which might, for example, explain the crash of 1987, when the US stock market lost 22.6% of its value in a single day). In the 1960s, Benoit Mandelbrot began to investigate whether Lévy processes described economic time series like cotton prices and stock prices. He found that the ones that generated jumps and extreme events better described financial markets.

It nearly, but not exactly, solved the problem of how to value an option—and thanks to Lefèvre, that means he came close to being able to value a complex portfolio of options, hedges, and speculations. The option pricing problem would not be solved precisely until much later in the twentieth century. The scholars who did so, Myron Scholes, Fischer Black, and Robert Merton, recognized Bachelier’s contribution. Scholes and Merton accepted the Nobel Prize in Economic Sciences in 1997 for their work on this important financial problem. Fischer Black had passed away before he could share in the award. MODELS AND MODERN MARKETS Scholes and Merton were professors of financial economics at the Massachusetts Institute of Technology in 1970, where they met the economist Fischer Black. Evidently none knew of Bachelier, and thus they had to retrace the mathematical logic of fair prices and random walks when they began work on the problem of option pricing in the late 1960s. Like Bachelier, they relied on a model of variation in prices—Brownian motion—although unlike Bachelier, they chose one that did not allow prices to become negative—a limitation of Bachelier’s work.


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Trading and Exchanges: Market Microstructure for Practitioners by Larry Harris

active measures, Andrei Shleifer, asset allocation, automated trading system, barriers to entry, Bernie Madoff, business cycle, buttonwood tree, buy and hold, compound rate of return, computerized trading, corporate governance, correlation coefficient, data acquisition, diversified portfolio, fault tolerance, financial innovation, financial intermediation, fixed income, floating exchange rates, High speed trading, index arbitrage, index fund, information asymmetry, information retrieval, interest rate swap, invention of the telegraph, job automation, law of one price, London Interbank Offered Rate, Long Term Capital Management, margin call, market bubble, market clearing, market design, market fragmentation, market friction, market microstructure, money market fund, Myron Scholes, Nick Leeson, open economy, passive investing, pattern recognition, Ponzi scheme, post-materialism, price discovery process, price discrimination, principal–agent problem, profit motive, race to the bottom, random walk, rent-seeking, risk tolerance, risk-adjusted returns, selection bias, shareholder value, short selling, Small Order Execution System, speech recognition, statistical arbitrage, statistical model, survivorship bias, the market place, transaction costs, two-sided market, winner-take-all economy, yield curve, zero-coupon bond, zero-sum game

After each flip, it is equal to its previous value plus 1 if the result is heads and minus 1 if the result is tails. Statisticians call this process a random walk because it describes the path a walker would take if after every step he flipped a coin to decide whether to next step forward or backward. Fully informative prices seem to follow random walks because no one can predict future price changes from past information when prices fully reflect that information. When price changes are unpredictable, they appear random. ◀ * * * Speculative arbitrages involve nonstationary hedge portfolios that arbitrageurs believe have a strong tendency toward short-term mean reversion. The nonstationariness is due to instrument-specific valuation factors that cause prices to follow a random walk in the long run. The mean reversion may come from inconsistent pricing of the common factors among the instruments in the hedge portfolio or from mispricing of one or more specific factors.

If fundamental value changes were predictable, current fundamental values would not fully reflect the information upon which the predictions are based. Fundamental value changes therefore must be unpredictable. Since prices are very close to fundamental values in efficient markets, price changes in efficient markets are quite unpredictable. When traders cannot predict future price changes, statisticians say that prices follow a random walk. Plots of random walks through time look like paths that wander up or down at random because random walks are completely unpredictable. * * * ▶ Fischer Black on Noise Fischer Black was a mathematician who made many seminal contributions to the development of financial theory. Perhaps most notably, he helped develop option-pricing theory, for which Myron Scholes and Robert Merton received the 1997 Nobel Prize in economic science. Had Fischer not died two years before the prize was awarded, he undoubtedly also would have been a Nobel laureate.

Unexpected changes in any of these factors generate fundamental volatility in the instrument. 2.1.2 Predictability Expected changes in fundamental factors generally do not change prices. Informative prices usually fully incorporate all available information about future values. Since people base their expectations on existing information, fully informative prices will already incorporate expected changes in fundamental factors. When the expected event occurs, it is not surprising, and it therefore should not cause prices to change. Only unexpected events cause fundamental price volatility. Consequently, the identifying characteristic of fundamental volatility in fully informative prices is unpredictable price changes. An unpredictable price process is called a random walk. Chapter 10 provides a more complete explanation of the properties of fully informative prices. * * * ▶ Gasoline, Diesel Fuel, and Heating Oil Volatility Gasoline, diesel fuel, and heating oil are expensive to store because they require very large tanks.


pages: 119 words: 10,356

Topics in Market Microstructure by Ilija I. Zovko

Brownian motion, computerized trading, continuous double auction, correlation coefficient, financial intermediation, Gini coefficient, information asymmetry, market design, market friction, market microstructure, Murray Gell-Mann, p-value, quantitative trading / quantitative finance, random walk, stochastic process, stochastic volatility, transaction costs

One of the predictions of the model, that to our knowledge has not been hypothesized elsewhere in the literature, is that the order size σ is an important determinant of the spread. Another prediction of the model concerns the price diffusion rate, which drives the volatility of prices and is the primary determinant of financial risk. If we assume that prices make a random walk, then the diffusion rate measures the size and frequency of its increments. The variance V of a random walk grows as V (t) = Dt, where D is the diffusion rate and t is time. This is the main free parameter in the Bachelier model of prices (Bachelier, 1964). While its value is essential for risk estimation and derivative pricing there is very little fundamental understanding of what actually determines it. In standard models it is often assumed to depend on “information arrival” (Clark, 1973), which has the disadvantage that it is impossible to measure directly.

We have done several studies, which will be reported in a future work, testing the importance of this effect. These show that while daily variations in W do give additional predictability for the spread, other aspects of the model are substantially responsible for these results. Measuring the price diffusion rate The measurement of the price diffusion rate requires some discussion. We measure the intraday price diffusion by computing the mid-point price variance V (τ ) = Var{m(t + τ ) − m(t)}, for different time scales τ . The averaging over t includes all events that change the mid-point price. The plot of V (τ ) against τ is called a diffusion curve and for an IID random walk is a straight line with slope D, the diffusion coefficient. 46 CHAPTER 3. THE PREDICTIVE POWER OF ZERO INTELLIGENCE IN FINANCIAL MARKETS 0.00008 0.00006 0.00004 Vodafone, August 4 1998 D=1.498e!06, R^2=0.998 0.00000 0.00002 <(m(t + τ) − m(t))2> 0.00010 0.00012 In our case, the computation of D is as follows: For each day we compute the diffusion curve.

One must bear in mind that the price diffusion rate from day to day has substantial correlations, as illustrated in Fig. 3.6. 0 20 40 60 80 τ Figure 3.5: Illustration of the procedure for measuring the price diffusion rate for Vodafone (VOD) on August 4th, 1998. On the x axis we plot the time τ in units of ticks, and on the y axis the variance of mid-price diffusion V (τ ). According to the hypothesis that mid-price diffusion is an uncorrelated Gaussian random walk, the plot should obey V (τ ) = Dτ . To cope with the fact that points with larger values of τ have fewer independent intervals and are less statistically significant, we use a weighted regression to compute the slope D. 47 !6.5 !7.5 !7.0 log(D) !6.0 !5.5 CHAPTER 3. THE PREDICTIVE POWER OF ZERO INTELLIGENCE IN FINANCIAL MARKETS 0 100 200 300 400 ACF 0.0 0.4 0.8 days 0 20 40 60 80 100 Lag Figure 3.6: Time series (top) and autocorrelation function (bottom) for daily price diffusion rate Dt for Vodafone. Because of longmemory effects and the short length of the series, the long-lag coefficients are poorly determined; the figure is just to demonstrate that the correlations are quite large. 3.5.6 Estimating the errors for the regressions The error bars presented in the text are based on a bootstrapping method.


pages: 354 words: 105,322

The Road to Ruin: The Global Elites' Secret Plan for the Next Financial Crisis by James Rickards

"Robert Solow", Affordable Care Act / Obamacare, Albert Einstein, asset allocation, asset-backed security, bank run, banking crisis, barriers to entry, Bayesian statistics, Ben Bernanke: helicopter money, Benoit Mandelbrot, Berlin Wall, Bernie Sanders, Big bang: deregulation of the City of London, bitcoin, Black Swan, blockchain, Bonfire of the Vanities, Bretton Woods, British Empire, business cycle, butterfly effect, buy and hold, capital controls, Capital in the Twenty-First Century by Thomas Piketty, Carmen Reinhart, cellular automata, cognitive bias, cognitive dissonance, complexity theory, Corn Laws, corporate governance, creative destruction, Credit Default Swap, cuban missile crisis, currency manipulation / currency intervention, currency peg, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, debt deflation, Deng Xiaoping, disintermediation, distributed ledger, diversification, diversified portfolio, Edward Lorenz: Chaos theory, Eugene Fama: efficient market hypothesis, failed state, Fall of the Berlin Wall, fiat currency, financial repression, fixed income, Flash crash, floating exchange rates, forward guidance, Fractional reserve banking, G4S, George Akerlof, global reserve currency, high net worth, Hyman Minsky, income inequality, information asymmetry, interest rate swap, Isaac Newton, jitney, John Meriwether, John von Neumann, Joseph Schumpeter, Kenneth Rogoff, labor-force participation, large denomination, liquidity trap, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, Mexican peso crisis / tequila crisis, money market fund, mutually assured destruction, Myron Scholes, Naomi Klein, nuclear winter, obamacare, offshore financial centre, Paul Samuelson, Peace of Westphalia, Pierre-Simon Laplace, plutocrats, Plutocrats, prediction markets, price anchoring, price stability, quantitative easing, RAND corporation, random walk, reserve currency, RFID, risk-adjusted returns, Ronald Reagan, Silicon Valley, sovereign wealth fund, special drawing rights, stocks for the long run, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, theory of mind, Thomas Bayes, Thomas Kuhn: the structure of scientific revolutions, too big to fail, transfer pricing, value at risk, Washington Consensus, Westphalian system

Starting from position 10, and following the walk represented by these random coin tosses, the position sequence is: 11 12 11 10 9 10 9 8 9. In this random walk, the walker moved 1 position (10 − 9 = 1) in 9 steps. This random sequence is referred to by scientists as disordered because the sequence does not show strong persistence in one direction or the other. If this experiment was repeated 1,000 times, easily done on a computer, the average distance from the starting place produced by the random walk of 9 steps would be approximately 3, which is the square root of 9. The distance of 3 = ta in our model. If t = 9 (the total steps taken), and ta ≈ 3 (total positions moved as shown by the random walk output), then a ≈ 0.5. The total movement in the 9-step random walk is 3 = 90.5. In a highly ordered walk a = 1.0. In a random or disordered walk a = 0.5.

With both in equal proportions the markets can function. What about the small minority of coin tossers? Their individual behavior is random. Do they cause markets as a whole to be random? Or do they cause bulls to become bears, and vice versa, producing nonrandom persistence? Research conducted by physicists Neil Johnson, Pak Ming Hui, and Paul Jefferies using financial market data shows the price movement pattern in markets does not correspond to the so-called random walk model that is the foundation of modern financial economics. Instead, behavior corresponds to predictions of complexity theorists using principles of feedback and adaptive behavior. Behavior in financial markets can be broken down into binary choices, expressed as “either/or” or “yes/no” answers to a series of questions. Will you trade stocks today? Will you consider IBM shares?

What type of walk do actual markets take? Stated formally, what is the value of a based on actual market price movements? One characteristic of complex systems is they are neither highly ordered nor random. Complex systems oscillate between order and disorder. This oscillation comes from agents’ deciding to quit the crowd and join the anticrowd or vice versa. A complex system that begins with random behavior can become ordered through feedback and adaptive behavior. Likewise, a highly ordered system can fall into disorder. Complex systems move back and forth, exactly as markets move from bull to bear phases as investor sentiment moves from fear to greed. Adaptation produces patterns more persistent than a random walk, tending toward order. Still, the system does not become completely ordered because of the crowd-anticrowd dynamic.


The Smartest Investment Book You'll Ever Read: The Simple, Stress-Free Way to Reach Your Investment Goals by Daniel R. Solin

asset allocation, buy and hold, corporate governance, diversification, diversified portfolio, index fund, market fundamentalism, money market fund, Myron Scholes, passive investing, prediction markets, random walk, risk tolerance, risk-adjusted returns, risk/return, transaction costs, Vanguard fund, zero-sum game

While a number of books have been written about the virtues of being a Smart Investor, few have achieved commercial 90 Your Broker or Advisor Is Keeping You from Being a Smart Investor success. One exception is A Random Walk Down Wall Street, a superb book by Bunon Malkiel, now in its eighth edition. Malkiel, a professor of econom ics at Princeton University, was one of the first to show that the history of the price of a stock cannot be used to predict how it will move in the future, and therefore that stock price movement is, in the language of economists. "random." In other words, he completely debunked the belief that anyone can consistently predict [he future prices of stocks (which is the core belief of Hyperactive Investors!). Most of the books and anicles written on the merits of being a Smarr Investor are, unfortunately, dense and difficult to understand-thus seemingly validating the myth that being a Smart Investor is somehow elitist, complex and beyond the ability of the ordinary investor.

See "Will Active Mutual Funds Continue [ 0 Underperform the Market in the Future?" by John Bogle. in Scott Simon's book Inda Mutual Funds: Profiting from an In mtmmt Rrvolution; see also the article by Edward S. O'Neal, discussed in Chapter 13. and a study by Dalbar, Inc .• a well~ respected research firm. Reported at http://www.dalbarinc.com/ con ten tIshowpage.asp ?page=200 1062 100. Burton Malkiel summarizes these studies in A Random Walk Down Wall Strut, p. 187. In Mark Hebner's book, Index Funds: Tht i2-Sup Program for Actiw InvtstorJ (pp. 47-53), he sets forth the studies showing the lack of consistency of mutual fund performance and the daunting odds of picking an actively managed fund that will outperform its benchmark index. One particularly compelling study referenced by Hebner indicated that, for the 1O~year period ending October 2004, oo1y 2.4% of the 1446 funds that had as a goal beating the S&P 500 Index succeeded in doing so.

The reprehensible conduct of hyperactive funds touting the "sizzle" of their past performance was recently exposed by the outstanding journalist Jonathan Clements of the Wall Street Journal in an online column entitled "Those PerformanceTouting Fund Ads Are Back-And That Could Mean Trouble." It is summarized at: http://socialize.morningstar.com/New Socialize/asp/FullConv.asp?forumId=F 1000000 15&lastConv Seq=41356 . Clements is the rare exception to those financial journalists who routinely peddle "financial pornography." Too Good 10 Be True? 153 Here is what Burton Malkid has to say about charting (wh ich he likens to "alchemy") in his sem inal book, A Random Walk Down Wall Strut, (p. 165): "There has been a rematkable unjformity in the conclusions of studies done on all forms of technical ana1ysis. Not one has consistently outperformed the placebo of a buy-and-hold strategy. Technical methods cannot be used to make useful investment strategies ... Ma1kiel believes that chartistS simply provide cover fo r hyperactive brokers to encourage more trading-generating more fees-by their unsuspecting clients.


pages: 335 words: 94,657

The Bogleheads' Guide to Investing by Taylor Larimore, Michael Leboeuf, Mel Lindauer

asset allocation, buy and hold, buy low sell high, corporate governance, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, Donald Trump, endowment effect, estate planning, financial independence, financial innovation, high net worth, index fund, late fees, Long Term Capital Management, loss aversion, Louis Bachelier, margin call, market bubble, mental accounting, money market fund, passive investing, Paul Samuelson, random walk, risk tolerance, risk/return, Sharpe ratio, statistical model, stocks for the long run, survivorship bias, the rule of 72, transaction costs, Vanguard fund, yield curve, zero-sum game

The study again found "no evidence of ability to predict successfully the direction of the stock market. " In the 1960s, a University of Chicago Professor, Eugene E Fama, performed a detailed analysis of the ever-increasing volume of stock price data. He concluded that stock prices are very efficient and that it's extremely difficult to pick winning stocks-especially after factoring in the costs of transaction fees. In 1973, Princeton professor Burton Malkiel, after extensive research, came to the same conclusion as Bachelier, Cowles, and Fama. Professor Malkiel published a book with the catchy title Random Walk Down Wall Street. The book is now an investment classic, and updated revisions are published on a regular basis. We think it deserves a place on the bookshelf of every serious investor. Professor Malkiel describes a random walk this way: "One in which future steps or directions cannot be predicted on the basis of past action.

When the term is applicable to the stock market, it means that short-run changes in stock prices cannot be predicted. " Another, more vivid, description of a random walk: A drunk standing in the middle of the road whose future movements can only be guessed. " Few academics argue that the stock market is totally efficient. Nevertheless, they agree that stocks and bonds are so efficiently priced that the majority of investors, including full-time professional fund managers, will not outperform an unmanaged index fund after transaction costs. Jack Bogle wrote: I know of no serious academic, professional money manager, trained security analyst, or intelligent individual investor who would disagree with the thrust of EMT The stock market itself is a demanding taskmaster. It sets a high hurdle that few investors can leap. Efficient markets and random walk are obscenities on Wall Street, where investors are constantly told that Wall Street's superior knowledge can make it easy to beat the market (for a fee).

Rebalancing may also improve your returns, since asset classes have had a tendency to revert to the mean (RTM) over time. By rebalancing, you're selling a portion of your winning asset classes before they revert to the mean (drop in price) and you're buying more of your underperforming asset classes when their prices are lower, before they revert to the mean (increase in value). So, you're selling high and buying low. If you believe in RTM, rebalancing could increase your returns. Jack Bogle believes in RTM, and we do, too. Even if you don't believe that RTM will occur in the future, but rather, believe that the market is a random walk and that each market move is independent of previous moves, remember that you'll still benefit from rebalancing because you're controlling the level of risk in your portfolio. Experienced investors have learned that risk control helps to keep your emotions in check and that in turn keeps your portfolio in line with your long-term plan.


Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

Black-Scholes formula, Brownian motion, buy and hold, buy low sell high, discrete time, fixed income, implied volatility, incomplete markets, martingale, random walk, short selling, stochastic process, stochastic volatility, transaction costs, volatility smile, Wiener process, zero-coupon bond

Proposition 2.24 Under the assumptions and notations of Definition 2.23, for all measurable deterministic functions F such that the corresponding random variables are integrable. Problem 2.25 Prove that a discrete time random walk is a Markov process. Vector processes Let ξ(t)=(ξ1(t),…, ξn(t)) be a vector process such that all its components are random processes. Then ξ is said to be an n-dimensional (vector) random process. All definitions given above can be extended for these vector processes. Sometimes, we can convert a process that is not a Markov process to a Markov process of higher dimension. Example 2.26 Let ηt be a random walk, t=0, 1, 2,…, and let Then ψt is not a Markov process, but the vector process (ηt,ψt) is a Markov process. 2.5 Problems Problem 2.27 Let ζ be a random variable, and let 0≤a<b≤1. Let a continuous time random process ξ(t) be such that Find the filtration when a=1/4, b=2/3

For simplicity, we shall use below stationary processes and white noise in the sense of Definitions 8.1–8.3, but all results are valid for wide-sense stationary processes and for the white noise defined as a wide-sense stationary process with no correlation and zero mean. 8.2 Simplest regression and autoregression The first-order regression model can be described by a one-dimensional equation © 2007 Nikolai Dokuchaev Review of Statistical Estimation 141 yt=β0+βxt+εt, t=1, 2,…. (8.1) Here yt and xt represent observable discrete time processes; yt is called the regressand, or dependent variable, xt is called the regressor, or explanatory variable, εt is an unobserved and are parameters that are usually unknown. error term, The standard assumption is that (8.2) Special case: autoregression (AR) Let us describe the first-order autoregressive process, AR(1), as yt=β0+βyt−1+εt, (8.3) where εt is a white noise process, are parameters. The AR(1) model is a special case of the simplest regression (8.1), where xt= yt−1. It can be shown that εt is uncorrelated with {ys}s<t. If −1<β<1, then there exists a stationary process such that as t→+∞. If β=1 and β0=0 in (8.3), then yt is a random walk (see Definition 2.6). A random walk is non-stationary and it does not converge to any stationary process. In fact, if |β|≥1, then Var yt→+∞ as t→+∞. This implies that many standard tools for forecasting and testing coefficients etc. are invalid. To avoid this, we can try to study changes in yt instead: for example, the differences zt=yt−yt−1 may converge to a stationary process. If not, the differences zt−zt−1 (i.e., the second differences yt−2yt−1+yt−2) may converge to a stationary process. 8.3 Least squares (LS) estimation Consider again the basic regression model (8.1)–(8.2).

Problem 8.40 Let yk=R(tk), where R(t)=ln S(tk), and where S(tk) are the daily prices for some stock, k=1, …, N, for some large enough N. (Find some prices on the internet or in newspapers.) (i) Using the LS estimator, forecast yk for N+2, N+10, N+100 days. Use this result to forecast the corresponding R(t). (ii) Estimate intervals that contain these increments of log of prices (and/or prices) with probability 0.7. © 2007 Nikolai Dokuchaev 9 Estimation of models for stock prices In this chapter, methods of statistical analysis are applied to historical stock prices. We show how to estimate the appreciation rate and the volatility for some continuous time stock price models. Some generic methods of forecast of evolution of prices and parameters are also given. 9.1 Review of the continuous time model Let us consider again the stock price equation dS(t)=a(t)S(t)dt+σ(t)S(t)dw(t), (9.1) where a(t) is the appreciation rate, σ(t) is the volatility, w(t) is a Wiener process, w(t)~N(0, t).


pages: 545 words: 137,789

How Markets Fail: The Logic of Economic Calamities by John Cassidy

"Robert Solow", Albert Einstein, Andrei Shleifer, anti-communist, asset allocation, asset-backed security, availability heuristic, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Black-Scholes formula, Blythe Masters, Bretton Woods, British Empire, business cycle, capital asset pricing model, centralized clearinghouse, collateralized debt obligation, Columbine, conceptual framework, Corn Laws, corporate raider, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, Daniel Kahneman / Amos Tversky, debt deflation, different worldview, diversification, Elliott wave, Eugene Fama: efficient market hypothesis, financial deregulation, financial innovation, Financial Instability Hypothesis, financial intermediation, full employment, George Akerlof, global supply chain, Gunnar Myrdal, Haight Ashbury, hiring and firing, Hyman Minsky, income per capita, incomplete markets, index fund, information asymmetry, Intergovernmental Panel on Climate Change (IPCC), invisible hand, John Nash: game theory, John von Neumann, Joseph Schumpeter, Kenneth Arrow, Kickstarter, laissez-faire capitalism, Landlord’s Game, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, mental accounting, Mikhail Gorbachev, money market fund, Mont Pelerin Society, moral hazard, mortgage debt, Myron Scholes, Naomi Klein, negative equity, Network effects, Nick Leeson, Northern Rock, paradox of thrift, Pareto efficiency, Paul Samuelson, Ponzi scheme, price discrimination, price stability, principal–agent problem, profit maximization, quantitative trading / quantitative finance, race to the bottom, Ralph Nader, RAND corporation, random walk, Renaissance Technologies, rent control, Richard Thaler, risk tolerance, risk-adjusted returns, road to serfdom, Robert Shiller, Robert Shiller, Ronald Coase, Ronald Reagan, shareholder value, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, statistical model, technology bubble, The Chicago School, The Great Moderation, The Market for Lemons, The Wealth of Nations by Adam Smith, too big to fail, transaction costs, unorthodox policies, value at risk, Vanguard fund, Vilfredo Pareto, wealth creators, zero-sum game

The coin-tossing model was resurrected: by the early 1960s, Samuelson and a number of other economists were publishing papers claiming that stock prices followed a random walk. One of these authors was Eugene Fama, an Italian American from Boston who was still in his early twenties. After paying his way through Tufts, Fama went to the University of Chicago, where he did his Ph.D. thesis on the behavior of stock prices, using the school’s spiffy new IBM mainframe to analyze data covering the period from 1926 to 1960. After providing a critical survey of previous research that had purported to find some predictability in stock returns, Fama reported details of his own statistical tests, which supported the random walk model. What made Fama’s paper especially distinctive was the criticism it contained of “fundamental analysis”—the type of stock research that many Wall Street professionals relied on, which involved deconstructing companies’ earnings reports, visiting factories, and so on.

(For many years, Malkiel served as a director of the Vanguard Group, which pioneered index funds. Fama joined another firm that manages index funds, Dimensional Fund Advisors.) The rise of efficient market theory also signaled the beginning of quantitative finance. In addition to the random walk model of stock prices, the period between 1950 and 1970 saw the development of the mean-variance approach to portfolio diversification, which Harry Markowitz, another Chicago economist, pioneered; the capital asset pricing model, which a number of different scholars developed independently of one another; and the Black-Scholes option pricing formula, which Fischer Black, an applied mathematician from Harvard, and Myron Scholes, a finance Ph.D. from Chicago, developed. Some of the mathematics used in these theories is pretty befuddling, which helps explain why there are so many physicists and mathematicians working on Wall Street, but the basic ideas underpinning them aren’t so difficult.

Posner, Richard Poulakakos, Harry Poulakakos, Peter Poundstone, William Prechter, Robert predictability, illusion of Prescott, Edward C. President’s Economic Policy Advisory Board Priceline Prices and Production (Hayek) Prince, Charles “Chuck” Princeton University Institute for Advanced Study Principles of Economics (Marshall) Principles of Political Economy (Mill) prisoner’s dilemma “Problem of Social Cost, The” (Coase) productivity agricultural growth of, random fluctuations in wages and Proud Decades, The (Diggins) Prudential Securities Quantum Fund Quarterly Journal of Economics, The Quesnay, François Rabin, Matt Radner, Roy Rajan, Raghuram G. Ramsey, Frank Rand, Ayn RAND Institute Random Walk Down Wall Street, A (Malkiel) random walk theory Ranieri, Lewis rational expectations theory RBS Greenwich Capital Reader’s Digest Reagan, Ronald reality-based economics RealtyTrac Reinhart, Vincent Renaissance Technologies “Report on Social Insurance and Allied Services” (Beveridge) Republican Party Reserve Primary Fund residential mortgage-backed securities (RMBSs) Resolution Trust Corporation Review of Economic Studies, The Revolution (Anderson) Revolutionary era Ricardo, David Rigas, John RiskMetrics Roach, Stephen S.


pages: 236 words: 77,735

Rigged Money: Beating Wall Street at Its Own Game by Lee Munson

affirmative action, asset allocation, backtesting, barriers to entry, Bernie Madoff, Bretton Woods, business cycle, buy and hold, buy low sell high, California gold rush, call centre, Credit Default Swap, diversification, diversified portfolio, estate planning, fiat currency, financial innovation, fixed income, Flash crash, follow your passion, German hyperinflation, High speed trading, housing crisis, index fund, joint-stock company, money market fund, moral hazard, Myron Scholes, passive investing, Ponzi scheme, price discovery process, random walk, risk tolerance, risk-adjusted returns, risk/return, stocks for the long run, stocks for the long term, too big to fail, trade route, Vanguard fund, walking around money

See national best bid and offer New York Stock Exchange Regulation (NYSE Regulation) no-transaction-fee funds (NTFs) noise non-correlated assets A Non-Random Walk Down Wall Street NTFs. See no-transaction-fee funds NYSE Regulation. See New York Stock Exchange Regulation O OER. See operating expense ratio OneSourse Select List operating cost operating expense ratio (OER) opinion, strong OPRA. See Options Price Reporting Authority Options Price Reporting Authority (OPRA) options strategies trading OTC. See Over-the-Counter Over-the-Counter (OTC) P Panic of 1907 passive investing Pay-Up Amendment. See Section 28(e) penny stocks pension pension manager Philip Morris pie charts bar charts versus Pit Bull play-it-safe investment portfolio, moderate risk premium price compression price discovery The Price Is Right price, best prime broker prognostication reports Q QQQ.

Usually used by advisers to impress clients with the broad diversification of terms a single asset class can be split up into. This aids in selling more mutual funds with different asset classes. For example, U.S. stocks can be split up into large-cap growth, large-cap value, large-cap high-dividend yield, and large-cap sector-that-is-currently-going-up which you don’t own because your sector is going down. The pitch continues that the classes do not move in tandem, but in a random walk unrelated to each other. A random walk is an overused term. Does anyone really think global markets are just walking around aimlessly with no rhyme or reason? Several people have been awarded the Nobel Prize in Economics for suggesting this. Perhaps the winners are chosen randomly as well. Of course, all of these asset classes are expected to go up over time. You wouldn’t buy something if it wasn’t designed to make a profit, right?

This is one reason many exchanges have circuit breakers to halt trading in a particular security or the whole exchange if prices go beyond a corridor or 5 to 10 percent in a short period of time. If prices move too fast, the risk increases that liquidity will disappear right at the moment it is needed the most. So, what changes the price? If you have more people who are willing to sell at the current market price than people who are willing to buy at a set limit, the price goes down. Think of it this way. Say you want to sell your house today for $100 (yes, you live in Detroit). If only one person walks by and offers $50, that’s the price you can sell for. If two people will pay $100, the price might rise until one backs down. In each case, the transaction price is based on what the other person is willing to pay; either a limit they set, or the market price you accept. When the market declines quickly, as we saw in the Flash Crash of May 6, 2010, many orders were placed to sell at the market price.


pages: 415 words: 125,089

Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein

"Robert Solow", Albert Einstein, Alvin Roth, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Bayesian statistics, Big bang: deregulation of the City of London, Bretton Woods, business cycle, buttonwood tree, buy and hold, capital asset pricing model, cognitive dissonance, computerized trading, Daniel Kahneman / Amos Tversky, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Lloyd's coffeehouse, endowment effect, experimental economics, fear of failure, Fellow of the Royal Society, Fermat's Last Theorem, financial deregulation, financial innovation, full employment, index fund, invention of movable type, Isaac Newton, John Nash: game theory, John von Neumann, Kenneth Arrow, linear programming, loss aversion, Louis Bachelier, mental accounting, moral hazard, Myron Scholes, Nash equilibrium, Norman Macrae, Paul Samuelson, Philip Mirowski, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Thaler, Robert Shiller, Robert Shiller, spectrum auction, statistical model, stocks for the long run, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, Thomas Bayes, trade route, transaction costs, tulip mania, Vanguard fund, zero-sum game

The normal distribution provides a more rigorous test of the random-walk hypothesis. But one qualification is important. Even if the random walk is a valid description of reality in the stock market-even if changes in stock prices fall into a perfect normal distribution-the mean will be something different from zero. The upward bias should come as no surprise. The wealth of owners of common stocks has risen over the long run as the economy and the revenues and profits of corporations have grown. Since more stock-price movements have been up than down, the average change in stock prices should work out to more than zero. In fact, the average increase in stock prices (excluding dividend income) was 7.7% a year. The standard deviation was 19.3%; if the future will resemble the past, this means that two-thirds of the time stock prices in any one year are likely to move within a range of +27.0% and -12.1%.

How closely do changes in the prices of stocks resemble a normal distribution? Some authorities on market behavior insist that stock prices follow a random walk-that they resemble the aimless and unplanned lurches of a drunk trying to grab hold of a lamppost. They believe that stock prices have no more memory than a roulette wheel or a pair of dice, and that each observation is independent of the preceding observation. Today's price move will be whatever it is going to be, regardless of what happened a minute ago, yesterday, the day before, or the day before that. The best way to determine whether changes in stock prices are in fact independent is to find out whether they fall into a normal distribution. Impressive evidence exists to support the case that changes in stock prices are normally distributed.

The charts in Chapter 8 (page 147) show that market performance over periods of a year or more does not look much like a normal distribution, but that performance by the month and by the quarter does, though not precisely. Quetelet would interpret that evidence as proof that stock-price movements in the short run are independent-that today's changes tell us nothing about what tomorrow's prices will be. The stock market is unpredictable. The notion of the random walk was evoked to explain why this should be so. But what about the longer view? After all, most investors, even impatient ones, stay in the market for more than a month, a quarter, or a year. Even though the contents of their portfolios change over time, serious investors tend to keep their money in the stock market for many years, even decades. Does the long run in the stock market really differ from the short run? If the random-walk view is correct, today's stock prices embody all relevant information. The only thing that would make them change is the availability of new information.


pages: 240 words: 60,660

Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

Albert Einstein, Asian financial crisis, Augustin-Louis Cauchy, Black-Scholes formula, British Empire, Brownian motion, capital asset pricing model, Cepheid variable, creative destruction, crony capitalism, diversified portfolio, Douglas Hofstadter, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Henri Poincaré, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, Isaac Newton, Johannes Kepler, law of one price, Mikhail Gorbachev, Myron Scholes, quantitative trading / quantitative finance, random walk, Richard Feynman, riskless arbitrage, savings glut, Schrödinger's Cat, Sharpe ratio, stochastic volatility, the scientific method, washing machines reduced drudgery, yield curve

(a) A single stock path simulated via a random walk. (b) Four typical simulated stock paths. (c) An actual four-year path for the level of the S&P 500 index. The EMM Isn’t Wild Enough Figure 5.4 compares the paths of stock prices generated from the EMM model with the level of the S&P 500 from late 2006 to late 2010. The apparently naïve either-up-or-down model does superficially mimic the riskiness of a stock’s price. But only more or less. The mimicry fails because the stock paths in the model are just too smooth when compared with the observed movements of actual stock prices. When examined closely, the stock price trajectory in Figure 5.4c is jerkier than those in Figure 5.4b. Actual stock prices are more wildly random than those of the model, as becomes very obvious during stock market crashes, when stock prices cascade downward in giant leaps, and volatility spikes beyond belief.

A MODEL FOR RISK Risk = The Uncertain Return on an Investment If you buy a stock for $100, you can imagine its price going up to $110 for a return of 10%, or down to $90 for a return of -10%. The risk of the stock is reflected in the range of possible returns you can envisage. A Random Walk for Stock Prices A company is a complex organism. How can one model the range of possible returns that a share of its stock might accrue? The Efficient Market Model’s answer to this question is radical: ignore complexity! It hypothesizes that the market, anthropomorphically speaking, has used all available knowledge about the company to determine the stock price. Therefore the next change in the stock price will arise only from new information, which will arrive randomly and therefore be equally likely to be good or bad as far as the company’s future returns are concerned.

The second assumption is pure model. The EMM’s picture of price movements goes by several names: a random walk, diffusion, and Brownian motion. One of its origins is in the description of the drift of pollen particles through a liquid as they collide with its molecules. Einstein used the diffusion model to successfully predict the square root of the average distance the pollen particles move through the liquid as a function of temperature and time, thus lending credence to the existence of hypothetical molecules and atoms too small to be seen. For particles of pollen, the model is also a theory, and pretty close to a true one. For stock prices, however, it’s only a model. It’s how we choose to imagine the way changes in stock prices occur, not what actually happens. It is naïve to imagine that the risk of every stock in the market can be condensed into just one quantity, its volatility σ.


Quantitative Trading: How to Build Your Own Algorithmic Trading Business by Ernie Chan

algorithmic trading, asset allocation, automated trading system, backtesting, Black Swan, Brownian motion, business continuity plan, buy and hold, compound rate of return, Edward Thorp, Elliott wave, endowment effect, fixed income, general-purpose programming language, index fund, John Markoff, Long Term Capital Management, loss aversion, p-value, paper trading, price discovery process, quantitative hedge fund, quantitative trading / quantitative finance, random walk, Ray Kurzweil, Renaissance Technologies, risk-adjusted returns, Sharpe ratio, short selling, statistical arbitrage, statistical model, survivorship bias, systematic trading, transaction costs

T 115 P1: JYS c07 JWBK321-Chan September 24, 2008 14:4 116 Printer: Yet to come QUANTITATIVE TRADING MEAN-REVERTING VERSUS MOMENTUM STRATEGIES Trading strategies can be profitable only if securities prices are either mean-reverting or trending. Otherwise, they are randomwalking, and trading will be futile. If you believe that prices are mean reverting and that they are currently low relative to some reference price, you should buy now and plan to sell higher later. However, if you believe the prices are trending and that they are currently low, you should (short) sell now and plan to buy at an even lower price later. The opposite is true if you believe prices are high. Academic research has indicated that stock prices are on average very close to random walking. However, this does not mean that under certain special conditions, they cannot exhibit some degree of mean reversion or trending behavior. Furthermore, at any given time, stock prices can be both mean reverting and trending depending on the time horizon you are interested in.

Unfortunately, most stock price series are not stationary—they exhibit a geometric random walk that gets them farther and farther away from their starting (i.e., initial public offering) values. However, you can often find P1: JYS c07 JWBK321-Chan September 24, 2008 14:4 Printer: Yet to come 127 Special Topics in Quantitative Trading a pair of stocks such that if you long one and short the other, the market value of the pair is stationary. If this is the case, then the two individual time series are said to be cointegrated. They are so described because a linear combination of them is integrated of order zero. Typically, two stocks that form a cointegrating pair are from the same industry group. Traders have long been familiar with this so-called pair-trading strategy. They buy the pair portfolio when the spread of the stock prices formed by these pairs is low, and sell/short the pair when the spread is high—in other words, a classic mean-reverting strategy.

Example 6.1: An Interesting Puzzle (or Why Risk Is Bad for You)* Here is a little puzzle that may stymie many a professional trader. Suppose a certain stock exhibits a true (geometric) random walk, by which I mean there is a 50–50 chance that the stock is going up 1 percent or down 1 percent every minute. If you buy this stock, are you most likely—in the long run and ignoring financing costs—to make money, lose money, or be flat? Most traders will blurt out the answer “Flat!,” and that is wrong. The correct answer is that you will lose money, at the rate of 0.005 percent (or 0.5 basis point) every minute! This is because for a geometric random walk, the average compounded rate of return is not the short-term (or one-period) return m (0 here), but is g = m − s 2 /2. This follows from the general formula for compounded growth g(f ) given in the appendix to this chapter, with the leverage f set to 1 and risk-free rate r set to 0.


pages: 923 words: 163,556

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures by Frank J. Fabozzi

algorithmic trading, Benoit Mandelbrot, capital asset pricing model, collateralized debt obligation, correlation coefficient, distributed generation, diversified portfolio, fixed income, index fund, Louis Bachelier, Myron Scholes, p-value, quantitative trading / quantitative finance, random walk, risk-adjusted returns, short selling, stochastic volatility, Thomas Bayes, transaction costs, value at risk

Random Walk Let us consider some price process given by the series {S}t.71 The dynamics of the process are given by(7.4) or, equivalently, ΔSt = εt . In words, tomorrow’s price, St+1, is thought of as today’s price plus some random shock that is independent of the price. As a consequence, in this model, known as the random walk, the increments St - St-1 from t−1 to t are thought of as completely undeterministic. Since the εt have a mean of zero, the increments are considered fair.72 An increase in price is as likely as a downside movement. At time t, the price is considered to contain all information available. So at any point in time, next period’s price is exposed to a random shock. Consequently, the best estimate for the following period’s price is this period’s price. Such price processes are called efficient due to their immediate information processing. A more general model, for example, AR(p), of the formSt = α0 + α1St −1 + … + αp St −p + εt with several lagged prices could be considered as well.

A more general model, for example, AR(p), of the formSt = α0 + α1St −1 + … + αp St −p + εt with several lagged prices could be considered as well. This price process would permit some slower incorporation of lagged prices into current prices. Now for the price to be a random walk process, the estimation would have to produce a0 = 0, a1 = 1, a2 = … = ap = 0. Application to S&P 500 Index Returns As an example to illustrate equation (7.4), consider the daily S&P 500 stock index prices between November 3, 2003 and December 31, 2003. The values are given in Table 7.2 along with the daily price changes. The resulting plot is given in Figure 7.4. The intuition given by the plot is roughly that, on each day, the information influencing the following day’s price is unpredictable and, hence, the price change seems completely arbitrary. Hence, at first glance much in this figure seems to support the concept of a random walk.

Hence, at first glance much in this figure seems to support the concept of a random walk. Concerning the evolution of the underling price process, it looks reasonable to assume that the next day’s price is determined by the previous day’s price plus some random change. From Figure 7.4, it looks as if the changes occur independently of each other and in a manner common to all changes (i.e., with identical distribution). Error Correction We next present a price model that builds on the relationship between spot and forward markets. Suppose we extend the random walk model slightly by introducing some forward price for the same underlying stock S. That is, at time t, we agree by contract to purchase the stock at t + 1 for some price determined at t. We denote this price by F(t). At time t + 1, we purchase the stock for F(t).


pages: 512 words: 162,977

New Market Wizards: Conversations With America's Top Traders by Jack D. Schwager

backtesting, beat the dealer, Benoit Mandelbrot, Berlin Wall, Black-Scholes formula, butterfly effect, buy and hold, commodity trading advisor, computerized trading, Edward Thorp, Elliott wave, fixed income, full employment, implied volatility, interest rate swap, Louis Bachelier, margin call, market clearing, market fundamentalism, money market fund, paper trading, pattern recognition, placebo effect, prediction markets, Ralph Nelson Elliott, random walk, risk tolerance, risk/return, Saturday Night Live, Sharpe ratio, the map is not the territory, transaction costs, War on Poverty

using both “Toward” and “Away From” motivation; having a goal of full capability plus, with anything less being unacceptable; breaking down potentially overwhelming goals into chunks, with satisfaction garnered from the completion of each individual step; keeping full concentration on the present moment—that is, the single task at hand rather than the long-term goal; being personally involved in achieving goals (as opposed to depending on others); and making self-to-self comparisons to measure progress. 41. PRICES ARE NONRANDOM = THE MARKETS CAN BE BEAT In reference to academicians who believe market prices are random, Trout says, “That’s probably why they’re professors and why Fin making money doing what I’m doing.” The debate over whether prices are random is not yet over. However, my experience with the interviews conducted for this book and its predecessor leaves me with little doubt that the random walk theory is wrong. It is not the magnitude of the winnings registered by the Market Wizards but the consistency of these winnings in some cases that underpin my belief. As a particularly compelling example, consider Blake’s 25:1 ratio of winning to losing months and his average annual return of 45 percent compared with a worst drawdown of only 5 percent.

And since cybernetic devices lack many of our human limitations, someday they’ll be able to do it better. I have no doubt that eventually the world’s best trader will be an automaton. I’m not saying this will happen soon, but probably within the next few generations. A good part of the academic community insists that the random nature of price behavior means that it’s impossible to develop trading systems that can beat the market over the long run. What’s your response? 124 / The New Market Wizard The evidence against the random walk theory of market action is staggering. Hundreds of traders and managers have profited from price-based mechanical systems. What about the argument that if you have enough people trading, some of them are going to do well, even if just because of chance? That may be true, but the probability of experiencing the kind of success that we have had and continue to have by chance alone has to be near zero.

Since then, there has been a steadily increasing number of papers providing evidence that the random walk theory is false. System trading has gone from a fringe idea to being a new kind of orthodoxy. I don’t think this could have happened if there weren’t something to it. However, I have to admit that I find it unsettling that what began as a renegade idea has become an element of the conventional wisdom. Of course, you can’t actually prove that price behavior is random. That’s right. You’re up against the problem of trying to prove a negative proposition. Although the contention that the markets are random is an affirmative proposition, in fact you’re trying to prove a negative. You’re trying to prove that there’s no systematic component in the price. Any negative proposition is very difficult to confirm because you’re trying to prove that something doesn’t exist.


pages: 416 words: 39,022

Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

asset allocation, Brownian motion, business continuity plan, business process, capital asset pricing model, computer age, corporate governance, discrete time, diversified portfolio, fixed income, implied volatility, index fund, interest rate derivative, iterative process, P = NP, p-value, random walk, risk/return, shareholder value, statistical model, stochastic process, transaction costs, value at risk, Wiener process, yield curve, zero-coupon bond

INTERNET SITES http://www.aptltd.com http://www.bis.org/index.htm http://www.cga-canada.org/fr/magazine/nov-dec02/Cyberguide f.htm http://www.fasb.org http://www.iasc.org.uk/cmt/0001.asp http://www.ifac.org http://www.prim.lu Index absolute global risk 285 absolute risk aversion coefficient 88 accounting standards 9–10 accrued interest 118–19 actuarial output rate on issue 116–17 actuarial return rate at given moment 117 adjustment tests 361 Aitken extrapolation 376 Akaike’s information criterion (AIC) 319 allocation independent allocation 288 joint allocation 289 of performance level 289–90 of systematic risk 288–9 American option 149 American pull 158–9 arbitrage 31 arbitrage models 138–9 with state variable 139–42 arbitrage pricing theory (APT) 97–8, 99 absolute global risk 285 analysis of style 291–2 beta 290, 291 factor-sensitivity profile 285 model 256, 285–94 relative global risk/tracking error 285–7 ARCH 320 ARCH-GARCH models 373 arithmetical mean 36–7 ARMA models 318–20 asset allocation 104, 274 asset liability management replicating portfolios 311–21 repricing schedules 301–11 simulations 300–1 structural risk analysis in 295–9 VaR in 301 autocorrelation test 46 autoregressive integrated moving average 320 autoregressive moving average (ARMA) 318 average deviation 41 bank offered rate (BOR) 305 basis point 127 Basle Committee for Banking Controls 4 Basle Committee on Banking Supervision 3–9 Basle II 5–9 Bayesian information criterion (BIC) 319 bear money spread 177 benchmark abacus 287–8 Bernouilli scheme 350 Best Linear Unbiased Estimators (BLUE) 363 beta APT 290, 291 portfolio 92 bijection 335 binomial distribution 350–1 binomial formula (Newton’s) 111, 351 binomial law of probability 165 binomial trees 110, 174 binomial trellis for underlying equity 162 bisection method 380 Black and Scholes model 33, 155, 174, 226, 228, 239 for call option 169 dividends and 173 for options on equities 168–73 sensitivity parameters 172–3 BLUE (Best Linear Unbiased Estimators) 363 bond portfolio management strategies 135–8 active strategy 137–8 duration and convexity of portfolio 135–6 immunizing a portfolio 136–7 positive strategy: immunisation 135–7 bonds average instant return on 140 390 Index bonds (continued ) definition 115–16 financial risk and 120–9 price 115 price approximation 126 return on 116–19 sources of risk 119–21 valuing 119 bootstrap method 233 Brennan and Schwarz model 139 building approach 316 bull money spread 177 business continuity plan (BCP) 14 insurance and 15–16 operational risk and 16 origin, definition and objective 14 butterfly money spread 177 calendar spread 177 call-associated bonds 120 call option 149, 151, 152 intrinsic value 153 premium breakdown 154 call–put parity relation 166 for European options 157–8 canonical analysis 369 canonical correlation analysis 307–9, 369–70 capital asset pricing model (CAPM or MEDAF) 93–8 equation 95–7, 100, 107, 181 cash 18 catastrophe scenarios 20, 32, 184, 227 Cauchy’s law 367 central limit theorem (CLT) 41, 183, 223, 348–9 Charisma 224 Chase Manhattan 224, 228 Choleski decomposition method 239 Choleski factorisation 220, 222, 336–7 chooser option 176 chord method 377–8 classic chord method 378 clean price 118 collateral management 18–19 compliance 24 compliance tests 361 compound Poisson process 355 conditional normality 203 confidence coefficient 360 confidence interval 360–1 continuous models 30, 108–9, 111–13, 131–2, 134 continuous random variables 341–2 contract-by-contract 314–16 convergence 375–6 convertible bonds 116 convexity 33, 149, 181 of a bond 127–9 corner portfolio 64 correlation 41–2, 346–7 counterparty 23 coupon (nominal) rate 116 coupons 115 covariance 41–2, 346–7 cover law of probability 164 Cox, Ingersoll and Ross model 139, 145–7, 174 Cox, Ross and Rubinstein binomial model 162–8 dividends and 168 one period 163–4 T periods 165–6 two periods 164–5 credit risk 12, 259 critical line algorithm 68–9 debentures 18 decision channels 104, 105 default risk 120 deficit constraint 90 degenerate random variable 341 delta 156, 181, 183 delta hedging 157, 172 derivatives 325–7 calculations 325–6 definition 325 extrema 326–7 geometric interpretations 325 determinist models 108–9 generalisation 109 stochastic model and 134–5 deterministic structure of interest rates 129–35 development models 30 diagonal model 70 direct costs 26 dirty price 118 discrete models 30, 108, 109–11. 130, 132–4 discrete random variables 340–1 dispersion index 26 distortion models 138 dividend discount model 104, 107–8 duration 33, 122–7, 149 and characteristics of a bond 124 definition 121 extension of concept of 148 interpretations 121–3 of equity funds 299 of specific bonds 123–4 Index dynamic interest-rate structure 132–4 dynamic models 30 dynamic spread 303–4 efficiency, concept of 45 efficient frontier 27, 54, 59, 60 for model with risk-free security 78–9 for reformulated problem 62 for restricted Markowitz model 68 for Sharpe’s simple index model 73 unrestricted and restricted 68 efficient portfolio 53, 54 EGARCH models 320, 373 elasticity, concept of 123 Elton, Gruber and Padberg method 79–85, 265, 269–74 adapting to VaR 270–1 cf VaR 271–4 maximising risk premium 269–70 equities definition 35 market efficiency 44–8 market return 39–40 portfolio risk 42–3 return on 35–8 return on a portfolio 38–9 security risk within a portfolio 43–4 equity capital adequacy ratio 4 equity dynamic models 108–13 equity portfolio diversification 51–93 model with risk-free security 75–9 portfolio size and 55–6 principles 515 equity portfolio management strategies 103–8 equity portfolio theory 183 equity valuation models 48–51 equivalence, principle of 117 ergodic estimator 40, 42 estimated variance–covariance matrix method (VC) 201, 202–16, 275, 276, 278 breakdown of financial assets 203–5 calculating VaR 209–16 hypotheses and limitations 235–7 installation and use 239–41 mapping cashflows with standard maturity dates 205–9 valuation models 237–9 estimator for mean of the population 360 European call 158–9 European option 149 event-based risks 32, 184 ex ante rate 117 ex ante tracking error 285, 287 ex post return rate 121 exchange options 174–5 exchange positions 204 391 exchange risk 12 exercise price of option 149 expected return 40 expected return risk 41, 43 expected value 26 exponential smoothing 318 extrema 326–7, 329–31 extreme value theory 230–4, 365–7 asymptotic results 365–7 attraction domains 366–7 calculation of VaR 233–4 exact result 365 extreme value theorem 230–1 generalisation 367 parameter estimation by regression 231–2 parameter estimation using the semi-parametric method 233, 234 factor-8 mimicking portfolio 290 factor-mimicking portfolios 290 factorial analysis 98 fair value 10 fat tail distribution 231 festoon effect 118, 119 final prediction error (FPE) 319 Financial Accounting Standards Board (FASB) 9 financial asset evaluation line 107 first derivative 325 Fisher’s skewness coefficient 345–6 fixed-income securities 204 fixed-rate bonds 115 fixed rates 301 floating-rate contracts 301 floating-rate integration method 311 FRAs 276 Fréchet’s law 366, 367 frequency 253 fundamental analysis 45 gamma 156, 173, 181, 183 gap 296–7, 298 GARCH models 203, 320 Garman–Kohlhagen formula 175 Gauss-Seidel method, nonlinear 381 generalised error distribution 353 generalised Pareto distribution 231 geometric Brownian motion 112, 174, 218, 237, 356 geometric mean 36 geometric series 123, 210, 328–9 global portfolio optimisation via VaR 274–83 generalisation of asset model 275–7 construction of optimal global portfolio 277–8 method 278–83 392 Index good practices 6 Gordon – Shapiro formula 48–50, 107, 149 government bonds 18 Greeks 155–7, 172, 181 gross performance level and risk withdrawal 290–1 Gumbel’s law 366, 367 models for bonds 149 static structure of 130–2 internal audit vs. risk management 22–3 internal notation (IN) 4 intrinsic value of option 153 Itô formula (Ito lemma) 140, 169, 357 Itô process 112, 356 Heath, Jarrow and Morton model 138, 302 hedging formula 172 Hessian matrix 330 high leverage effect 257 Hill’s estimator 233 historical simulation 201, 224–34, 265 basic methodology 224–30 calculations 239 data 238–9 extreme value theory 230–4 hypotheses and limitations 235–7 installation and use 239–41 isolated asset case 224–5 portfolio case 225–6 risk factor case 224 synthesis 226–30 valuation models 237–8 historical volatility 155 histories 199 Ho and Lee model 138 homogeneity tests 361 Hull and White model 302, 303 hypothesis test 361–2 Jensen index 102–3 Johnson distributions 215 joint allocation 289 joint distribution function 342 IAS standards 10 IASB (International Accounting Standards Board) 9 IFAC (International Federation of Accountants) 9 immunisation of bonds 124–5 implied volatility 155 in the money 153, 154 independence tests 361 independent allocation 288 independent random variables 342–3 index funds 103 indifference curves 89 indifference, relation of 86 indirect costs 26 inequalities on calls and puts 159–60 inferential statistics 359–62 estimation 360–1 sampling 359–60 sampling distribution 359–60 instant term interest rate 131 integrated risk management 22, 24–5 interest rate curves 129 kappa see vega kurtosis coefficient 182, 189, 345–6 Lagrangian function 56, 57, 61, 63, 267, 331 for risk-free security model 76 for Sharpe’s simple index model 71 Lagrangian multipliers 57, 331 law of large numbers 223, 224, 344 law of probability 339 least square method 363 legal risk 11, 21, 23–4 Lego approach 316 leptokurtic distribution 41, 182, 183, 189, 218, 345 linear equation system 335–6 linear model 32, 33, 184 linearity condition 202, 203 Lipschitz’s condition 375–6 liquidity bed 316 liquidity crisis 17 liquidity preference 316 liquidity risk 12, 16, 18, 296–7 logarithmic return 37 logistic regression 309–10, 371 log-normal distribution 349–50 log-normal law with parameter 349 long (short) straddle 176 loss distribution approach 13 lottery bonds 116 MacLaurin development 275, 276 mapping cashflows 205–9 according to RiskMetricsT M 206–7 alternative 207–8 elementary 205–6 marginal utility 87 market efficiency 44–8 market model 91–3 market price of the risk 141 market risk 12 market straight line 94 Index market timing 104–7 Markowitz’s portfolio theory 30, 41, 43, 56–69, 93, 94, 182 first formulation 56–60 reformulating the problem 60–9 mathematic valuation models 199 matrix algebra 239 calculus 332–7 diagonal 333 n-order 332 operations 333–4 symmetrical 332–3, 334–5 maturity price of bond 115 maximum outflow 17–18 mean 343–4 mean variance 27, 265 for equities 149 measurement theory 344 media risk 12 Merton model 139, 141–2 minimum equity capital requirements 4 modern portfolio theory (MPT) 265 modified duration 121 money spread 177 monoperiodic models 30 Monte Carlo simulation 201, 216–23, 265, 303 calculations 239 data 238–9 estimation method 218–23 hypotheses and limitations 235–7 installation and use 239–41 probability theory and 216–18 synthesis 221–3 valuation models 237–8 multi-index models 221, 266 multi-normal distribution 349 multivariate random variables 342–3 mutual support 147–9 Nelson and Schaefer model 139 net present value (NPV) 298–9, 302–3 neutral risk 164, 174 New Agreement 4, 5 Newson–Raphson nonlinear iterative method 309, 379–80, 381 Newton’s binomial formula 111, 351 nominal rate of a bond 115, 116 nominal value of a bond 115 non-correlation 347 nonlinear equation systems 380–1 first-order methods 377–9 iterative methods 375–7 n-dimensional iteration 381 principal methods 381 393 solving 375–81 nonlinear Gauss-Seidel method 381 nonlinear models independent of time 33 nonlinear regression 234 non-quantifiable risks 12–13 normal distribution 41, 183, 188–90, 237, 254, 347–8 normal law 188 normal probability law 183 normality 202, 203, 252–4 observed distribution 254 operational risk 12–14 business continuity plan (BCP) and 16 definition 6 management 12–13 philosophy of 5–9 triptych 14 options complex 175–7 definition 149 on bonds 174 sensitivity parameters 155–7 simple 175 strategies on 175–7 uses 150–2 value of 153–60 order of convergence 376 Ornstein – Uhlenbeck process 142–5, 356 OTC derivatives market 18 out of the money 153, 154 outliers 241 Pareto distribution 189, 367 Parsen CAT 319 partial derivatives 329–31 payment and settlement systems 18 Pearson distribution system 183 perfect market 31, 44 performance evaluation 99–108 perpetual bond 123–4 Picard’s iteration 268, 271, 274, 280, 375, 376, 381 pip 247 pockets of inefficiency 47 Poisson distribution 350 Poisson process 354–5 Poisson’s law 351 portfolio beta 92 portfolio risk management investment strategy 258 method 257–64 risk framework 258–64 power of the test 362 precautionary surveillance 3, 4–5 preference, relation of 86 394 Index premium 149 price at issue 115 price-earning ratio 50–1 price of a bond 127 price variation risk 12 probability theory 216–18 process risk 24 product risk 23 pseudo-random numbers 217 put option 149, 152 quadratic form 334–7 qualitative approach 13 quantifiable risks 12, 13 quantile 188, 339–40 quantitative approach 13 Ramaswamy and Sundaresan model 139 random aspect of financial assets 30 random numbers 217 random variables 339–47 random walk 45, 111, 203, 355 statistical tests for 46 range forwards 177 rate fluctuation risk 120 rate mismatches 297–8 rate risk 12, 303–11 redemption price of bond 115 regression line 363 regressions 318, 362–4 multiple 363–4 nonlinear 364 simple 362–3 regular falsi method 378–9 relative fund risk 287–8 relative global risk 285–7 relative risks 43 replicating portfolios 302, 303, 311–21 with optimal value method 316–21 repos market 18 repricing schedules 301–11 residual risk 285 restricted Markowitz model 63–5 rho 157, 173, 183 Richard model 139 risk, attitude towards 87–9 risk aversion 87, 88 risk factors 31, 184 risk-free security 75–9 risk, generalising concept 184 risk indicators 8 risk management cost of 25–6 environment 7 function, purpose of 11 methodology 19–21 vs back office 22 risk mapping 8 risk measurement 8, 41 risk-neutral probability 162, 164 risk neutrality 87 risk of one equity 41 risk of realisation 120 risk of reinvestment 120 risk of reputation 21 risk per share 181–4 risk premium 88 risk return 26–7 risk transfer 14 risk typology 12–19 Risk$TM 224, 228 RiskMetricsTM 202, 203, 206–7, 235, 236, 238, 239–40 scenarios and stress testing 20 Schaefer and Schwartz model 139 Schwarz criterion 319 scope of competence 21 scorecards method 7, 13 security 63–5 security market line 107 self-assessment 7 semi-form of efficiency hypothesis 46 semi-parametric method 233 semi-variance 41 sensitivity coefficient 121 separation theorem 94–5, 106 series 328 Sharpe’s multi-index model 74–5 Sharpe’s simple index method 69–75, 100–1, 132, 191, 213, 265–9 adapting critical line algorithm to VaR 267–8 cf VaR 269 for equities 221 problem of minimisation 266–7 VaR in 266–9 short sale 59 short-term interest rate 130 sign test 46 simulation tests for technical analysis methods 46 simulations 300–1 skewed distribution 182 skewness coefficient 182, 345–6 specific risk 91, 285 speculation bubbles 47 spot 247 Index spot price 150 spot rate 129, 130 spreads 176–7 square root process 145 St Petersburg paradox 85 standard Brownian motion 33, 355 standard deviation 41, 344–5 standard maturity dates 205–9 standard normal law 348 static models 30 static spread 303–4 stationarity condition 202, 203, 236 stationary point 327, 330 stationary random model 33 stochastic bond dynamic models 138–48 stochastic differential 356–7 stochastic duration 121, 147–8 random evolution of rates 147 stochastic integral 356–7 stochastic models 109–13 stochastic process 33, 353–7 particular 354–6 path of 354 stock exchange indexes 39 stock picking 104, 275 stop criteria 376–7 stop loss 258–9 straddles 175, 176 strangles 175, 176 strategic risk 21 stress testing 20, 21, 223 strike 149 strike price 150 strong form of efficiency hypothesis 46–7 Student distribution 189, 235, 351–2 Student’s law 367 Supervisors, role of 8 survival period 17–18 systematic inefficiency 47 systematic risk 44, 91, 285 allocation of 288–9 tail parameter 231 taste for risk 87 Taylor development 33, 125, 214, 216, 275–6 Taylor formula 37, 126, 132, 327–8, 331 technical analysis 45 temporal aspect of financial assets 30 term interest rate 129, 130 theorem of expected utility 86 theoretical reasoning 218 theta 156, 173, 183 three-equity portfolio 54 395 time value of option 153, 154 total risk 43 tracking errors 103, 285–7 transaction risk 23–4 transition bonds 116 trend extrapolations 318 Treynor index 102 two-equity portfolio 51–4 unbiased estimator 360 underlying equity 149 uniform distribution 352 uniform random variable 217 utility function 85–7 utility of return 85 utility theory 85–90, 183 valuation models 30, 31–3, 160–75, 184 value at risk (VaR) 13, 20–1 based on density function 186 based on distribution function 185 bond portfolio case 250–2 breaking down 193–5 calculating 209–16 calculations 244–52 component 195 components of 195 definition 195–6 estimation 199–200 for a portfolio 190–7 for a portfolio of linear values 211–13 for a portfolio of nonlinear values 214–16 for an isolated asset 185–90 for equities 213–14 heading investment 196–7 incremental 195–7 individual 194 link to Sharp index 197 marginal 194–5 maximum, for portfolio 263–4 normal distribution 188–90 Treasury portfolio case 244–9 typology 200–2 value of basis point (VBP) 19–20, 21, 127, 245–7, 260–3 variable contracts 301 variable interest rates 300–1 variable rate bonds 115 variance 41, 344–5 variance of expected returns approach 183 variance – covariance matrix 336 Vasicek model 139, 142–4, 174 396 Index vega (kappa) 156, 173 volatility of option 154–5 yield curve 129 yield to maturity (YTM) 250 weak form of the efficiency hypothesis 46 Weibull’s law 366, 367 Wiener process 355 zero-coupon bond 115, 123, 129 zero-coupon rates, analysis of correlations on 305–7 Index compiled by Annette Musker

Bonds 141 By differentiating the value of the portfolio, we have: dVt = −Pt (s1 )(µt (s1 ) dt − σt (s1 ) dwt ) + X · Pt (s2 )(µt (s2 ) dt − σt (s2 ) dwt ) = [−Pt (s1 )µt (s1 ) + XPt (s2 )µt (s2 )] · dt + [Pt (s1 )σt (s1 ) − XPt (s2 )σt (s2 )] · dwt The arbitrage logic will therefore lead us to  −Pt (s1 )µt (s1 ) + XPt (s2 )µt (s2 )   = rt  −P (s ) + XP (s ) t t 1 2  P (s )σ (s ) − XPt (s2 )σt (s2 )   t 1 t 1 =0 −Pt (s1 ) + XPt (s2 ) In other words: XPt (s2 ) · (µt (s2 ) − rt ) = Pt (s1 ) · (µt (s1 ) − rt ) XPt (s2 ) · σt (s2 ) = Pt (s1 ) · σt (s1 ) We can eliminate X, for example by dividing the two equations member by member, which gives: µt (s1 ) − rt µt (s2 ) − rt = σt (s1 ) σt (s2 ) This shows that the expression λt (rt ) = µt (s) − rt is independent of s; this expression σt (s) is known as the market price of the risk. By replacing µt and σt with their value in the preceding relation, we arrive at Pt + (a + λb)Pr + b2 P − rP = 0 2 rr What we are looking at here is the partial derivatives equation of the second order, which together with the initial condition Ps (s, rt ) = l, defines the price process. This equation must be resolved for each specification of a(t, rt ), b(t, rt ) and λt (rt ). 4.5.1.2 The Merton model31 Because of its historical interest,32 we are showing the simplest model, the Merton model. This model assumes that the instant term rate follows a random walk model: drt = α · dt + σ · dwt with α and σ being constant and the market price of risk being zero (λ = 0). The partial derivatives equation for the prices takes the form: Pt + αPr + σ 2 P − rP = 0. 2 rr 31 Merton R., Theory of rational option pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141–83. 32 This is in fact the first model based on representation of changes in the spot rate using a stochastic differential equation. 142 Asset and Risk Management It is easy to verify that the solution to this equation (with the initial condition) is given by Pt (s, rt ) = exp −(s − t)rt − α σ2 (s − t)2 + (s − t)3 2 6 The average instant return rate is given by µt (s, rt ) = Pt + αPr + P σ 2 P 2 rr = rt · P = r t P which shows that in this case, the average return is independent of the maturity date.

This question is addressed in the following paragraphs, and the analysis is carried out at three levels according to the accessibility of information. The least that can be said is that the conclusions of the searches carried out in order to test efficiency are inconclusive and should not be used as a basis for forming clear and definitive ideas. 9 Fama E. F., Behaviour of Stock Market Prices, Journal of Business, Vol. 38, 1965, pp. 34–105. Fama E. F., Random Walks in Stock Market Prices, Financial Analysis Journal, 1965. Fama E. F., Efficient Capital Markets: A Review of Theory and Empirical Work, Journal of Finance, Vol. 25, 1970. 10 This approach is adopted in this work. 11 Refer for example to Bechu T. and Bertrand E., L’Analyse Technique, Economica, 1998. 46 Asset and Risk Management 3.1.2.2 Weak form The weak form of the efficiency hypothesis postulates that it is not possible to gain a particular advantage from the range of historical observations; the rates therefore purely and simply include the previous rate values.


pages: 403 words: 119,206

Toward Rational Exuberance: The Evolution of the Modern Stock Market by B. Mark Smith

bank run, banking crisis, business climate, business cycle, buy and hold, capital asset pricing model, compound rate of return, computerized trading, credit crunch, cuban missile crisis, discounted cash flows, diversified portfolio, Donald Trump, Eugene Fama: efficient market hypothesis, financial independence, financial innovation, fixed income, full employment, income inequality, index arbitrage, index fund, joint-stock company, locking in a profit, Long Term Capital Management, Louis Bachelier, margin call, market clearing, merger arbitrage, money market fund, Myron Scholes, Paul Samuelson, price stability, random walk, Richard Thaler, risk tolerance, Robert Bork, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, stocks for the long run, the market place, transaction costs

In such an efficient market, securities are instantaneously priced to reflect all information available to market participants. It is therefore theoretically impossible to predict future price moves based on current publicly available information. In effect, Fama confirmed the prescient insight the young French mathematician Louis Bachelier first developed in 1900. In an efficient market, no one could really hope to “beat” the market, because the market already perfectly reflected the sum total of all relevant data. But there was more. Nine months after publishing the article in the Journal of Business, Fama wrote a simplified version for the Financial Analysts Journal entitled “Random Walks in Stock Market Prices.” Comparing the movement of stock prices to the “random walk” of a drunk stumbling from point to point, Fama argued that price movements in an “efficient market” were random, representing adjustments to unpredictable news items that, when made public, would immediately be reflected in the price of stocks.

., & Company Pier One Polaroid Pond Coal Company pools; bankers’; outlawed Porter, Sylvia portfolios; “baby,” ; diversification of; of high-yield bonds; insurance of; liquidation of, during bear markets; mutual fund; pension plan; “replicating,” ; risk-efficient Pratt, Serano preferred stock present value, determination of Price, T. Rowe, Jr. Price, William W. price-earnings (P/E) ratios; during boom of 1920s; and crash of 1929; and crash of 1962; interest rates and; of IBM; of Nifty Fifty; postwar; during recession-depression of 1920–21; during recession of 1970s; rise in, during 1950s; of S&P; during World War I; during World War II Princeton University probability theory Procter & Gamble Produce Exchange productivity, increase in program trading progressive politics Prohibition Prudential Prudent Man Rule Public Broadcasting System (PBS) Pulitzer, Joseph Putnam, Samuel Radio Corporation of America (RCA) railroads; bankruptcy of; see also specific railroad companies “random walk” concept Reagan, Ronald recessions; of 1920–21; of 1950s; of 1970s; of 1982 Remington Rand Republican Party repurchases retirement plans; regulation of Revenue Act (1942) Revlon Rinfret, Pierre risk arbitrage risk-reward relationships “robber barons,” Robbins, Lionel Roberts, Harry Robinson, James Harvey Rockefeller, John D.

The determination of these fluctuations depends on an infinite number of factors; it is, therefore, impossible to aspire to mathematical predictions of it … the dynamics of the Exchange will never be an exact science.9 Bachelier had another important insight—that stock price fluctuations tend to grow larger as the time horizon lengthens. The formula he developed to describe the phenomenon bears a remarkable resemblance to the formula that describes the random collision of molecules as they move in space. Many years later this process would be described as a random walk, a key concept underlying much of the academic work on the stock market in the second half of the twentieth century. A great deal of Bachelier’s work was revolutionary. He laid the groundwork upon which later mathematicians constructed a full-fledged theory of probability, and made the first theoretical attempts to value options and futures. All this was done in an effort to explain why stock prices were impossible to predict.10 Bachelier was not modest about his work.


Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

Black-Scholes formula, Brownian motion, capital asset pricing model, cellular automata, delta neutral, discounted cash flows, discrete time, diversified portfolio, fixed income, interest rate derivative, interest rate swap, locking in a profit, London Interbank Offered Rate, margin call, martingale, quantitative trading / quantitative finance, random walk, short selling, stochastic process, time value of money, transaction costs, value at risk, Wiener process, zero-coupon bond

Glossary of Symbols A B β c C C CA CE CE Cov delta div div0 D D DA E E∗ f F gamma Φ k K i m fixed income (risk free) security price; money market account bond price beta factor covariance call price; coupon value covariance matrix American call price European call price discounted European call price covariance Greek parameter delta dividend present value of dividends derivative security price; duration discounted derivative security price price of an American type derivative security expectation risk-neutral expectation futures price; payoff of an option; forward rate forward price; future value; face value Greek parameter gamma cumulative binomial distribution logarithmic return return coupon rate compounding frequency; expected logarithmic return 305 306 Mathematics for Finance M m µ N N k ω Ω p p∗ P PA PE PE PA r rdiv re rF rho ρ S S σ t T τ theta u V Var VaR vega w w W x X y z market portfolio expected returns as a row matrix expected return cumulative normal distribution the number of k-element combinations out of N elements scenario probability space branching probability in a binomial tree risk-neutral probability put price; principal American put price European put price discounted European put price present value factor of an annuity interest rate dividend yield effective rate risk-free return Greek parameter rho correlation risky security (stock) price discounted risky security (stock) price standard deviation; risk; volatility current time maturity time; expiry time; exercise time; delivery time time step Greek parameter theta row matrix with all entries 1 portfolio value; forward contract value, futures contract value variance value at risk Greek parameter vega symmetric random walk; weights in a portfolio weights in a portfolio as a row matrix Wiener process, Brownian motion position in a risky security strike price position in a fixed income (risk free) security; yield of a bond position in a derivative security Index admissible – portfolio 5 – strategy 79, 88 American – call option 147 – derivative security – put option 147 amortised loan 30 annuity 29 arbitrage 7 at the money 169 attainable – portfolio 107 – set 107 183 basis – of a forward contract 128 – of a futures contract 140 basis point 218 bear spread 208 beta factor 121 binomial – distribution 57, 180 – tree model 7, 55, 81, 174, 238 Black–Derman–Toy model 260 Black–Scholes – equation 198 – formula 188 bond – at par 42, 249 – callable 255 – face value 39 – fixed-coupon 255 – floating-coupon 255 – maturity date 39 – stripped 230 – unit 39 – with coupons 41 – zero-coupon 39 Brownian motion 69 bull spread 208 butterfly 208 – reversed 209 call option 13, 181 – American 147 – European 147, 188 callable bond 255 cap 258 Capital Asset Pricing Model 118 capital market line 118 caplet 258 CAPM 118 Central Limit Theorem 70 characteristic line 120 compounding – continuous 32 – discrete 25 – equivalent 36 – periodic 25 – preferable 36 conditional expectation 62 contingent claim 18, 85, 148 – American 183 – European 173 continuous compounding 32 continuous time limit 66 correlation coefficient 99 coupon bond 41 coupon rate 249 307 308 covariance matrix 107 Cox–Ingersoll–Ross model 260 Cox–Ross–Rubinstein formula 181 cum-dividend price 292 delta 174, 192, 193, 197 delta hedging 192 delta neutral portfolio 192 delta-gamma hedging 199 delta-gamma neutral portfolio 198 delta-vega hedging 200 delta-vega neutral portfolio 198 derivative security 18, 85, 253 – American 183 – European 173 discount factor 24, 27, 33 discounted stock price 63 discounted value 24, 27 discrete compounding 25 distribution – binomial 57, 180 – log normal 71, 186 – normal 70, 186 diversifiable risk 122 dividend yield 131 divisibility 4, 74, 76, 87 duration 222 dynamic hedging 226 effective rate 36 efficient – frontier 115 – portfolio 115 equivalent compounding 36 European – call option 147, 181, 188 – derivative security 173 – put option 147, 181, 189 ex-coupon price 248 ex-dividend price 292 exercise – price 13, 147 – time 13, 147 expected return 10, 53, 97, 108 expiry time 147 face value 39 fixed interest 255 fixed-coupon bond 255 flat term structure 229 floating interest 255 floating-coupon bond 255 floor 259 floorlet 259 Mathematics for Finance forward – contract 11, 125 – price 11, 125 – rate 233 fundamental theorem of asset pricing 83, 88 future value 22, 25 futures – contract 134 – price 134 gamma 197 Girsanov theorem 187 Greek parameters 197 growth factor 22, 25, 32 Heath–Jarrow–Morton model hedging – delta 192 – delta-gamma 199 – delta-vega 200 – dynamic 226 in the money 169 initial – forward rate 232 – margin 135 – term structure 229 instantaneous forward rate interest – compounded 25, 32 – fixed 255 – floating 255 – simple 22 – variable 255 interest rate 22 interest rate option 254 interest rate swap 255 261 233 LIBID 232 LIBOR 232 line of best fit 120 liquidity 4, 74, 77, 87 log normal distribution 71, 186 logarithmic return 34, 52 long forward position 11, 125 maintenance margin 135 margin call 135 market portfolio 119 market price of risk 212 marking to market 134 Markowitz bullet 113 martingale 63, 83 Index 309 martingale probability 63, 250 maturity date 39 minimum variance – line 109 – portfolio 108 money market 43, 235 no-arbitrage principle 7, 79, 88 normal distribution 70, 186 option – American 183 – at the money 169 – call 13, 147, 181, 188 – European 173, 181 – in the money 169 – interest rate 254 – intrinsic value 169 – out of the money 169 – payoff 173 – put 18, 147, 181, 189 – time value 170 out of the money 169 par, bond trading at 42, 249 payoff 148, 173 periodic compounding 25 perpetuity 24, 30 portfolio 76, 87 – admissible 5 – attainable 107 – delta neutral 192 – delta-gamma neutral 198 – delta-vega neutral 198 – expected return 108 – market 119 – variance 108 – vega neutral 197 positive part 148 predictable strategy 77, 88 preferable compounding 36 present value 24, 27 principal 22 put option 18, 181 – American 147 – European 147, 189 put-call parity 150 – estimates 153 random interest rates random walk 67 rate – coupon 249 – effective 36 237 – forward 233 – – initial 232 – – instantaneous 233 – of interest 22 – of return 1, 49 – spot 229 regression line 120 residual random variable 121 residual variance 122 return 1, 49 – expected 53 – including dividends 50 – logarithmic 34, 52 reversed butterfly 209 rho 197 risk 10, 91 – diversifiable 122 – market price of 212 – systematic 122 – undiversifiable 122 risk premium 119, 123 risk-neutral – expectation 60, 83 – market 60 – probability 60, 83, 250 scenario 47 security market line 123 self-financing strategy 76, 88 short forward position 11, 125 short rate 235 short selling 5, 74, 77, 87 simple interest 22 spot rate 229 Standard and Poor Index 141 state 238 stochastic calculus 71, 185 stochastic differential equation 71 stock index 141 stock price 47 strategy 76, 87 – admissible 79, 88 – predictable 77, 88 – self-financing 76, 88 – value of 76, 87 strike price 13, 147 stripped bond 230 swap 256 swaption 258 systematic risk 122 term structure 229 theta 197 time value of money 21 310 trinomial tree model Mathematics for Finance 64 underlying 85, 147 undiversifiable risk 122 unit bond 39 value at risk 202 value of a portfolio 2 value of a strategy 76, 87 VaR 202 variable interest 255 Vasiček model 260 vega 197 vega neutral portfolio volatility 71 weights in a portfolio Wiener process 69 yield 216 yield to maturity 229 zero-coupon bond 39 197 94

Next, S(nτ + τ ) 1 ≈ 1 + mτ + σξ(n + 1) + σ 2 τ S(nτ ) 2 1 2 = 1 + m + σ τ + σξ(n + 1), 2 and so S(nτ + τ ) − S(nτ ) ≈ 1 m + σ 2 S(nτ )τ + σS(nτ )ξ(n + 1). 2 Since ξ(n + 1) = w(nτ + τ ) − w(nτ ), we obtain an approximate equation describing the dynamics of stock prices: S(t + τ ) − S(t) ≈ 1 m + σ 2 S(t)τ + σS(t)(w(t + τ ) − w(t)), 2 (3.8) where t = nτ . The solution S(t) of this approximate equation is given by the same formula as in Proposition 3.7. For any N = 1, 2, . . . we consider a binomial tree model with time step of length τ = N1 . Let SN (t) be the corresponding stock prices and let wN (t) be the corresponding symmetric random walk with increments ξN (t) = wN (t) − n is the time after n steps. wN (t − N1 ), where t = N Exercise 3.25 Compute the expectation and variance of wN (t), where t = n N. 3. Risky Assets 69 We shall use the Central Limit Theorem2 to obtain the limit as N → ∞ of the random walk wN (t). To this end we put x(n) = k(n) − mτ √ σ τ for each n = 1, 2, . . . , which is a sequence of independent identically distributed random variables, each with expectation 0 and variance 1.

Risky Assets 67 Introducing a sequence of independent random variables ξ(n), each with two values √ + τ with probability 1/2, √ ξ(n) = − τ with probability 1/2, we can write the logarithmic return as k(n) = mτ + σξ(n). Exercise 3.23 Find the expectation and variance of ξ(n) and k(n). Exercise 3.24 Write S(1) and S(2) in terms of m, σ, τ , ξ(1) and ξ(2). Next, we introduce an important sequence of random variables w(n), called a symmetric random walk, such that w(n) = ξ(1) + ξ(2) + · · · + ξ(n), and w(0) = 0. Clearly, ξ(n) = w(n) − w(n − 1). Because of the last equality, the ξ(n) are referred to as the increments of w(n). From now on we shall often write S(t) and w(t) instead of S(n) and w(n) for t = τ n, where n = 1, 2, . . . . Proposition 3.7 The stock price at time t = τ n is given by S(t) = S(0) exp(mt + σw(t)). Proof By (3.2) S(t) = S(nτ ) = S(nτ − τ )ek(n) = S(nτ − 2τ )ek(n−1)+k(n) = · · · = S(0)ek(1)+···+k(n) = S(0)emnτ +σ(ξ(1)+···+ξ(n)) = S(0)emt+σw(t) , as required. 68 Mathematics for Finance In order to pass to the continuous-time limit we use the approximation 1 ex ≈ 1 + x + x2 , 2 accurate for small values of x, to obtain S(nτ + τ ) 1 = ek(n+1) ≈ 1 + k(n + 1) + k(n + 1)2 .


High-Frequency Trading by David Easley, Marcos López de Prado, Maureen O'Hara

algorithmic trading, asset allocation, backtesting, Brownian motion, capital asset pricing model, computer vision, continuous double auction, dark matter, discrete time, finite state, fixed income, Flash crash, High speed trading, index arbitrage, information asymmetry, interest rate swap, latency arbitrage, margin call, market design, market fragmentation, market fundamentalism, market microstructure, martingale, natural language processing, offshore financial centre, pattern recognition, price discovery process, price discrimination, price stability, quantitative trading / quantitative finance, random walk, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, Tobin tax, transaction costs, two-sided market, yield curve

The opportunity cost is the sum of these returns weighted by the 188 i i i i i i “Easley” — 2013/10/8 — 11:31 — page 189 — #209 i i IMPLEMENTATION SHORTFALL WITH TRANSITORY PRICE EFFECTS size of the unexecuted orders. While opportunity costs are a real concern for institutional investors, our methodology does not offer much insight into them, and in the rest of the chapter we focus only on the execution cost component. OBSERVED PRICES, EFFICIENT PRICES AND PRICING ERRORS The implementation shortfall incorporates the total price impact of a large order. However, to better understand the sources of the shortfall, it may be useful to decompose the price impact into its permanent and transitory components. To do this we must define and measure the efficient price and any deviations from it at each moment in time. We take the standard approach of assuming the efficient price is unpredictable, ie, it follows a random walk. Minus trading frictions, the efficient price at the daily or intra-day frequency can be characterised as a martingale process.

Minus trading frictions, the efficient price at the daily or intra-day frequency can be characterised as a martingale process. Let mj be this latent price (9.7) mj = mj−1 + wt Sometimes the quote midpoint is assumed to represent this latent price. However, quote midpoints are not generally martingales with respect to all available order flow, in which case Hasbrouck (1995, p. 1179) proposes to view the random-walk component of a Stock and Watson (1988) decomposition as the “implicit efficient price”. Hasbrouck (2007, Chapters 4 and 8) constructs an efficient price more generally as the projection of mt onto all available conditioning variables, ie, the so-called filtered state estimate m̃ij = E∗ [mj | pij , pi,j−1 , . . . ] (9.8) where E∗ [·] is the linear projection of mij on a set of lagged prices.4 A standard approach to implementing such a projection is through autoregressive integrated moving average (ARIMA) time series econometrics (Hasbrouck 2007, Chapter 4).

Ané and Geman (2000) is another notable, more recent contribution. Mandelbrot and Taylor open their paper with the following assertion: Price changes over a fixed number of transactions may have a Gaussian distribution. Price changes over a fixed time period may follow a stable Paretian distribution, whose variance is infinite. 5 i i i i i i “Easley” — 2013/10/8 — 11:31 — page 6 — #26 i i HIGH-FREQUENCY TRADING Since the number of transactions in any time period is random, the above statements are not necessarily in disagreement.… Basically, our point is this: the Gaussian random walk as applied to transactions is compatible with a symmetric stable Paretian random walk as applied to fixed time intervals. In other words, Mandelbrot and Taylor advocated for recovering normality through a transaction-based clock, moving away from chronological time.


pages: 345 words: 87,745

The Power of Passive Investing: More Wealth With Less Work by Richard A. Ferri

asset allocation, backtesting, Bernie Madoff, buy and hold, capital asset pricing model, cognitive dissonance, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, endowment effect, estate planning, Eugene Fama: efficient market hypothesis, fixed income, implied volatility, index fund, intangible asset, Long Term Capital Management, money market fund, passive investing, Paul Samuelson, Ponzi scheme, prediction markets, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, survivorship bias, too big to fail, transaction costs, Vanguard fund, yield curve, zero-sum game

Fama’s meticulously researched Ph.D. thesis was published in 1965 and titled “The Behavior of Stock Market Prices.” The purpose of the paper was to test the theory that stock market prices are random and follow what’s commonly referred to today as a random walk.8 Fama’s work led to the formation of the efficient market hypothesis (EMH), which is a theory of efficient security pricing in free and open markets. The theory states that all known and available information is already reflected in current securities prices. Thus, the price agreed to by a willing buyer and seller in the open market is the best estimate, good or bad, of the investment value of a security. Any new information is nearly instantaneously incorporated into market prices. This makes it almost impossible to capture excess returns without taking greater risk or having inside information about securities.

See Early performance studies Permanent loss Persistence of performance academic studies: bond funds Carhart’s work Fama and French “Hot Hands” study Personal trust(s): fiduciary investing and taxes and Pioneering Portfolio Management (Swensen) Plain vanilla index Policy changes Ponzi scheme Poor accounting Portfolio choices: bottom line and changing the model efficient portfolios fund selection strategies modeling the active bet modifications to model portfolios of active funds quantifying of random portfolio results real-world test relative performance model short-term/long-term Portfolio management: annual evaluation debate on facts about objective of options for Portfolio Selection: Efficient Diversification of Investments (Markowitz) Portfolio theory, modern Positive period weighting Predictors of top performance: fund expenses as qualitative factors as ratings as Pre-inflation return Price-earnings ratio (P/E): growth/value stocks portfolio returns and Price-to-book (P/B) Price-to-cash-flow Price Waterhouse Private trust management: categories of trusts restatement of trusts (third) taxes and UPIA and active management UPIA and passive investing Procrastinating non-index investors: changing/staying the course definition of endowment effect and land of the lost modern portfolio theory and veering off course Prospect theory Prudence, elements of Prudent Investor Act: A Guide to Understanding, The (Simon) Prudent Investor Rule Prudent Man Rule “Purity Hypothesis, The” Qualitative factors, performance and Random walk Random Walk Down Wall Street, A (Malkiel) Rating methods, performance and Real estate Real Estate Investment Trust Act Real Estate Investment Trusts (REITs) Real return Rebalancing portfolio Recovery, market Registered investment advisor (RIA) Reinganum, Marc REITs.

“Portfolio Selection,” The Journal of Finance 7, no. 1 (March 1952): 77–91. 8. Eugene Fama, “The Behavior of Stock Market Prices,” Journal of Business 38, no. 1 (January 1965): 34–105. 9. Ibid., 92. 10. William F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” The Journal of Finance 19, no. 3 (1964): 425–42. 11. Jack L. Treynor, “How to Rate Management of Investment Funds,” Harvard Business Review 43 (1965): 63–75. 12. William F. Sharpe, “Mutual Fund Performance,” Journal of Business 39 (1996): 119–138. 13. Michael C. Jensen, “The Performance of Mutual Funds in the Period 1945–1964,” The Journal of Finance 23, no. 2 (1967): 389–416. Chapter 3: The Birth of Index Funds 1. Burton Malkiel, A Random Walk Down Wall Street (New York: W.W. Norton, 1973). 2. Paul A. Samuelson, “Challenge to Judgment,” Journal of Portfolio Management 1, no. 1 (1974): 17–19. 3.


Data Mining: Concepts and Techniques: Concepts and Techniques by Jiawei Han, Micheline Kamber, Jian Pei

bioinformatics, business intelligence, business process, Claude Shannon: information theory, cloud computing, computer vision, correlation coefficient, cyber-physical system, database schema, discrete time, distributed generation, finite state, information retrieval, iterative process, knowledge worker, linked data, natural language processing, Netflix Prize, Occam's razor, pattern recognition, performance metric, phenotype, random walk, recommendation engine, RFID, semantic web, sentiment analysis, speech recognition, statistical model, stochastic process, supply-chain management, text mining, thinkpad, Thomas Bayes, web application

The eccentricity of a is 2, that is, , , and . Thus, the radius of G is 2, and the diameter is 3. Note that it is not necessary that . Vertices c, d, and e are peripheral vertices. Figure 11.13 A graph, G, where vertices c, d, and e are peripheral. SimRank: Similarity Based on Random Walk and Structural Context For some applications, geodesic distance may be inappropriate in measuring the similarity between vertices in a graph. Here we introduce SimRank, a similarity measure based on random walk and on the structural context of the graph. In mathematics, a random walk is a trajectory that consists of taking successive random steps. Similarity between people in a social network Let's consider measuring the similarity between two vertices in the AllElectronics customer social network of Example 11.18. Here, similarity can be explained as the closeness between two participants in the network, that is, how close two people are in terms of the relationship represented by the social network.

Ada and Bob may randomly forward such information to their friends (or neighbors ) in the network. The closeness between Ada and Bob can then be measured by the likelihood that other customers simultaneously receive the promotional information that was originally sent to Ada and Bob. This kind of similarity is based on the random walk reachability over the network, and thus is referred to as similarity based on random walk. Let's have a closer look at what is meant by similarity based on structural context, and similarity based on random walk. The intuition behind similarity based on structural context is that two vertices in a graph are similar if they are connected to similar vertices. To measure such similarity, we need to define the notion of individual neighborhood. In a directed graph , where V is the set of vertices and is the set of edges, for a vertex , the individual in-neighborhood of v is defined as(11.29) Symmetrically, we define the individual out-neighborhood of v as(11.30) Following the intuition illustrated in Example 11.20, we define SimRank, a structural-context similarity, with a value that is between 0 and 1 for any pair of vertices.

The probability of the tour is defined as(11.35) To measure the probability that a vertex w receives a message that originated simultaneously from u and v, we extend the expected distance to the notion of expected meeting distance, that is,(11.36) where is a pair of tours and of the same length. Using a constant C between 0 and 1, we define the expected meeting probability as(11.37) which is a similarity measure based on random walk. Here, the parameter C specifies the probability of continuing the walk at each step of the trajectory. It has been shown that for any two vertices, u and v. That is, SimRank is based on both structural context and random walk. 11.3.3. Graph Clustering Methods Let's consider how to conduct clustering on a graph. We first describe the intuition behind graph clustering. We then discuss two general categories of graph clustering methods. To find clusters in a graph, imagine cutting the graph into pieces, each piece being a cluster, such that the vertices within a cluster are well connected and the vertices in different clusters are connected in a much weaker way.


pages: 263 words: 75,455

Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors by Wesley R. Gray, Tobias E. Carlisle

activist fund / activist shareholder / activist investor, Albert Einstein, Andrei Shleifer, asset allocation, Atul Gawande, backtesting, beat the dealer, Black Swan, business cycle, butter production in bangladesh, buy and hold, capital asset pricing model, Checklist Manifesto, cognitive bias, compound rate of return, corporate governance, correlation coefficient, credit crunch, Daniel Kahneman / Amos Tversky, discounted cash flows, Edward Thorp, Eugene Fama: efficient market hypothesis, forensic accounting, hindsight bias, intangible asset, Louis Bachelier, p-value, passive investing, performance metric, quantitative hedge fund, random walk, Richard Thaler, risk-adjusted returns, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, statistical model, survivorship bias, systematic trading, The Myth of the Rational Market, time value of money, transaction costs

In a collection of essays called The Random Character of Stock Market Prices (1964), Thorp read the English translation of a French dissertation written in 1900 by a student at the University of Paris, Louis Bachelier. Bachelier's dissertation unlocked the secret to valuing warrants: the so-called “random walk” theory. As the name suggests, the “random walk” holds that the movements made by security prices are random. While it might seem paradoxical, the random nature of the moves makes it possible to probabilistically determine the future price of the security. The implications of the random walk theory are profound, and they weren't lost on Thorp. He saw that he could apply the theory to handicap the value of the warrant. Where the warrant's price differed from Thorp's probabilistic valuation, Thorp recognized that an opportunity existed for him to trade the warrant and the underlying stock and to profit from the differential.

See Look-ahead bias Price ratios analysis of compound annual growth rates alpha and adjusted performance risk-adjusted performance and absolute measures of risk value premium and spread book-to-market composite formed from all metrics formed from the “best” price ratios top-performing earnings yield EBIT variation, outperformance by enterprise yield (EBITDA and EBIT variations) forward earnings estimate free cash flow yield gross profits yield long-term study methods of studying Princeton-Newport Partners PROBM model Procter & Gamble Profit margins growth maximum stability Pronovost, Peter Puthenpurackal, John Quality and Price, improving compared with Magic Formula finding Price finding Quality Quantitative value checklist Quantitative value strategy examining, results of analysis legend beating the market black box, looking inside man versus machine risk and return robustness Greenblatt's Magic Formula bargain price examination of findings good business Quality and Price, improving compared with Magic Formula finding Price finding Quality simplifying strategy implementation checklist tried-and-true value investing principles Quinn, Kevin The Random Character of Stock Market Prices (Bachelier) Random walk theory Regression analysis Representativeness heuristic “Returns to Trading Strategies Based on Price-to-Earnings and Price-to-Sales Ratios” (Nathan, Sivakumar, & Vijayakumar) Ridgeline Partners Risk-adjusted performance and absolute measures of risk R-squared Ruane, William Scaled net operating assets (SNOA) Scaled total accruals (STA) Schedule 13D Security Analysis (Graham & Dodd) See's Candies Self-attribution bias Sequoia Fund Sharpe, William Sharpe ratio Shiller, Robert Short selling Shumway, Tyler Simons, Jim Singleton, Henry Sloan, Richard Small sample bias “Some Insiders Are Indeed Smart Investors” (Giamouridis, Liodakis, & Moniz) Sortino ratio Stock buybacks, issuance, and announcements Stock market, predicting movements in sustainable alpha quantitative value strategy simplifying tried-and-true value investing principles model, testing benchmarking data errors historical data versus forward data size of portfolio and target stocks small sample bias transaction costs universe, parameters of Super Crunchers: Why Thinking-by-Numbers Is the New Way to Be Smart (Ayres) “The Superinvestors of Graham-and-Doddsville” (Buffett) Survivorship bias Sustainable alpha Taleb, Nassim Teledyne Tetlock, Philip Theory of Investment Value (Williams) Third Avenue Value Fund Thorp, Ed Total enterprise value (TEV) Transaction costs Tsai, Claire Tversky, Amos Value investors'errors Value portfolio Value premium and spread Wellman, Jay What Works on Wall Street (O'Shaughnessy) Whitman, Martin J.

They agreed, however, on one very important point: both believed it was possible to outperform the stock market, a belief that flew in the face of the efficient market hypothesis. While it is true that Thorp's strategy was grounded in the random walk, a key component of the efficient market hypothesis, he disagreed with the efficient market believers that it necessarily implied that markets were efficient. Indeed, Thorp went so as far as to call his book Beat the Market. Buffett also thought the efficient market hypothesis was nonsense, writing in his 1988 Shareholder Letter15: This doctrine [the efficient market hypothesis] became highly fashionable—indeed, almost holy scripture in academic circles during the 1970s. Essentially, it said that analyzing stocks was useless because all public information about them was appropriately reflected in their prices. In other words, the market always knew everything. As a corollary, the professors who taught EMT said that someone throwing darts at the stock tables could select a stock portfolio having prospects just as good as one selected by the brightest, most hard-working security analyst.


pages: 267 words: 71,941

How to Predict the Unpredictable by William Poundstone

accounting loophole / creative accounting, Albert Einstein, Bernie Madoff, Brownian motion, business cycle, butter production in bangladesh, buy and hold, buy low sell high, call centre, centre right, Claude Shannon: information theory, computer age, crowdsourcing, Daniel Kahneman / Amos Tversky, Edward Thorp, Firefox, fixed income, forensic accounting, high net worth, index card, index fund, John von Neumann, market bubble, money market fund, pattern recognition, Paul Samuelson, Ponzi scheme, prediction markets, random walk, Richard Thaler, risk-adjusted returns, Robert Shiller, Robert Shiller, Rubik’s Cube, statistical model, Steven Pinker, transaction costs

These are the lowest prices you could have obtained on the Web at any given moment in 2012. Though erratic, the price is clearly not a random walk. That means it’s predictable to a degree. The average rock-bottom Web price for this Xbox bundle was $379 (about £230). The price hovered close to that average for most of the year. You could have paid as little as $280 (£170) or as much as $600 (£370). The chart’s main feature is the sharp increase on September 1, followed by a decline to Black Friday. This is a not-uncommon pattern for holiday gift items. Retailers raise the price so that it can be lowered for holiday “sales.” The Xbox price sank to a favorable level for Black Friday and Cyber Monday, and then went abruptly up to $462 (£280). It would have been unwise to buy at that price. Bargain hunters shop on Black Friday and a few days afterwards.

Son of a policeman, Niederhoffer took an economics PhD at the University of Chicago, the bastion of the random walk theory, which holds that short-term changes in stock prices are completely unpredictable. “I criticized all those who had concluded that markets were random, including most of the professors in the room,” Niederhoffer said. “Further, I cautioned them that their failure to disprove a hypothesis … was methodologically inadequate to support a conclusion that prices were random. When I put it in the vernacular, ‘You can’t prove a negative,’ pandemonium broke loose.” Niederhoffer collaborated with M.F.M. Osborne on a paper that could be called the Magna Carta of high-frequency trading. Osborne was another outsider to the economics profession, an astrophysicist who worked for a navy think tank. He had done important work on the random walk hypothesis. But in “Market Making and Reversal on the Stock Exchange,” published in the Journal of the American Statistical Association (1966), Niederhoffer and Osborne argued that stock price movements are not random at all.

See hot hand theory presidential campaign (1964), 89 price-to-earnings (PE) ratio. See stock market price-to-earnings (PE) ratios priming, 54, 56 probabilities, 79, 81, 87–88, 167–172 Proceedings of the American Philosophical Society, 109, 110 Procter & Gamble, 207 product placement, 28 property prices, 199–204 Psotka, Joseph, 103–105 psychic experiments. See ESP (extrasensory perception)/telepathy publicity stunts, 27–29 pupil dilation, 89–90 “Purloined Letter, The” (Poe), 10, 54–55 push-button keypads (for telephones), 39–40 radio, 27–33 random walk theory, 213 randomness, 171–172, 183–184, 250–251 crowd-sourced ratings and, 105 deck of cards and, 164–165 difficulty in achieving, 4 ESP machine and, 39 experiments by Chapanis, 40–44 experiments, history of, 43–45 fake numbers and, 114, 120, 122–123, 125 hot hand theory and, 158–160, 162–169 human perceptions of, 22 Lacan and, 55 lotteries and, 72, 74–75 mentalists and, 45–49 passwords and, 94, 97–98, 101, 102 in playing card games, 86–90 random vs. random-looking, 37 rock/paper/scissors (RPS) and, 55 soccer penalty kicks and, 83–85 stock market and, 213–214 tennis and, 79–81 Zenith experiments and, 35–38 See also outguessing machines; tests ratings.


Risk Management in Trading by Davis Edwards

asset allocation, asset-backed security, backtesting, Black-Scholes formula, Brownian motion, business cycle, computerized trading, correlation coefficient, Credit Default Swap, discrete time, diversified portfolio, fixed income, implied volatility, intangible asset, interest rate swap, iterative process, John Meriwether, London Whale, Long Term Capital Management, margin call, Myron Scholes, Nick Leeson, p-value, paper trading, pattern recognition, random walk, risk tolerance, risk/return, selection bias, shareholder value, Sharpe ratio, short selling, statistical arbitrage, statistical model, stochastic process, systematic trading, time value of money, transaction costs, value at risk, Wiener process, zero-coupon bond

FIGURE 3.8 (+) Leptokurtic (0) Mesokurtic (Normal) (–) Platykurtic Kurtosis RANDOM WALKS (STOCHASTIC PROCESSES) A random walk is a special type of random process that describes the path taken by a series of random steps. In finance, random walk processes are commonly used to model how prices or interest rates might move in the future. In finance, most models are usually limited to a single dimension (like an interest rate rising and falling) rather than a more general case (like a model of a gas particle, which can move in three dimensions). Mathematically, a random process is usually described as the current value being equal to the previous value plus some random change in value. The change in value is commonly represented by the capital Greek letter Delta, Δ. (See Equation 3.5, A Simple Random Process.) Simulating price of some asset, abbreviated P: Pt = Pt −1 + ΔP (3.5) where Pt Price at time t.

Develop random inputs Generate a simulation using the random inputs Calculate the result of trading with the strategy Run the simulation multiple times and aggregate the results. Develop Random Inputs Monte Carlo simulations are driven by random inputs. A common choice for an input is a model that adds random movement onto an existing price (this is called stochastic process). For example, gold prices might be modeled assuming they start at the current price (observed in the market today) and then follow a random walk into the future where each day a random adjustment is applied to the price from the previous day. These inputs are typically a professional judgment—and not all model inputs will work out equally well in real life. To minimize some of the judgment, inputs are typically calibrated by analyzing historical data. This helps guarantee that a model at least matches historical observations.

These assumptions are 208 RISK MANAGEMENT IN TRADING made in order to enable an easy‐to‐use framework suitable for pricing a wide variety of options. The key assumptions made by Black Scholes models: 1. Arbitrage‐Free Markets. Most option models, including Black Scholes, assume that traders try to maximize their personal profits and don’t allow arbitrage opportunities (riskless opportunities to make a profit) to persist. An implication of this assumption is no one can predict the future. Anyone who can predict the future would already have made trades, and kept borrowing money to make trades, until prices change enough to remove the potential for profit. This makes future price movements a random walk where prices are equally likely to move up or down around a time‐ value of money adjusted for fair value. 2. Frictionless, Continuous Markets.


pages: 586 words: 159,901

Wall Street: How It Works And for Whom by Doug Henwood

accounting loophole / creative accounting, activist fund / activist shareholder / activist investor, affirmative action, Andrei Shleifer, asset allocation, asset-backed security, bank run, banking crisis, barriers to entry, borderless world, Bretton Woods, British Empire, business cycle, capital asset pricing model, capital controls, central bank independence, computerized trading, corporate governance, corporate raider, correlation coefficient, correlation does not imply causation, credit crunch, currency manipulation / currency intervention, David Ricardo: comparative advantage, debt deflation, declining real wages, deindustrialization, dematerialisation, diversification, diversified portfolio, Donald Trump, equity premium, Eugene Fama: efficient market hypothesis, experimental subject, facts on the ground, financial deregulation, financial innovation, Financial Instability Hypothesis, floating exchange rates, full employment, George Akerlof, George Gilder, hiring and firing, Hyman Minsky, implied volatility, index arbitrage, index fund, information asymmetry, interest rate swap, Internet Archive, invisible hand, Irwin Jacobs, Isaac Newton, joint-stock company, Joseph Schumpeter, kremlinology, labor-force participation, late capitalism, law of one price, liberal capitalism, liquidationism / Banker’s doctrine / the Treasury view, London Interbank Offered Rate, Louis Bachelier, market bubble, Mexican peso crisis / tequila crisis, microcredit, minimum wage unemployment, money market fund, moral hazard, mortgage debt, mortgage tax deduction, Myron Scholes, oil shock, Paul Samuelson, payday loans, pension reform, plutocrats, Plutocrats, price mechanism, price stability, prisoner's dilemma, profit maximization, publication bias, Ralph Nader, random walk, reserve currency, Richard Thaler, risk tolerance, Robert Gordon, Robert Shiller, Robert Shiller, selection bias, shareholder value, short selling, Slavoj Žižek, South Sea Bubble, The inhabitant of London could order by telephone, sipping his morning tea in bed, the various products of the whole earth, The Market for Lemons, The Nature of the Firm, The Predators' Ball, The Wealth of Nations by Adam Smith, transaction costs, transcontinental railway, women in the workforce, yield curve, zero-coupon bond

Fama, Eugene F. (1965a). "The Behavior of Stock Prices," Journal of Business 57, pp. 34-105. — (1965b). "Random Walks in Stock Market Prices," Financial Analysts JournaKSeptem- ber-October), pp. 55-59. — (1968). "What 'Random Walk' Really Means," Institutional Investor (April), pp. 38-40. — (1970). "Efficient Capital Markets: A Review of Theory and Empirical ^ork," Journal of Finance 25. pp. 383-423. — (1980). "Banking in the Theory of Finance,"/owmfl/ of Monetary Economics^, pp. 3S>-57. — (1981). "Stock Returns, Real Activity, Inflation, and Money," American Economic Re- view 71, pp. 545-565. — (1991). "Efficient Capital Markets: II," Journal of Finance 46, pp. 1575-1617. Fama, Eugene F., and Kenneth R. French (1988). "Permanent and Temporary Components of Stock Prices," Journal of Political Economy 96, pp. 246-273- — (1989) "Business Conditions and Expected Returns on Stocks and Bonds,"/owrna/ of Financial Economics 25, pp. 23^9

Work in the 1980s, using different — a skeptic on econometrics hesitates to use the phrase "more powerful" — statistical MARKET MODELS techniques, repeatedly found more patterns in prices than classic random walk theory allowed, especially if you examine portfolios consisting of a number of stocks. In other words, individual shares may follow more-or-less random patterns, but the market as a whole moves in trends (or, in statistical language, returns are autocorrelated), especially over the long term. "[Rjecent research," Fama concluded, "is able to show confidently that daily and weekly returns are predictable from past returns." Even more challenging to EMH orthodoxy is research that goes beyond price history. A number of studies during the 1970s and 1980s showed that stocks with low price/earnings ratios and/or high dividend yields — "cheap" stocks, in slang — would likely outperform stocks that were expensive by these measures, a sophisticated confirmation of old Wall Street wisdom.

Others, little better than haruspices, try to divine patterns in price graphs that supposedly portend dramatic upward or downward moves. Such "chartists" speak enthusiastically of pennants, rising wedges, head and shoulders, saucer bottoms. There is little evidence that chart-reading works at all; the patterns seen are probably little different from the butterflies and genitalia that one sees in a Rorschach test. The economist Burton Malkiel, author of the popular investment text A Random Walk Down Wall Street, had his students construct mythical stock price charts by flipping coins. When Malkiel showed these to practicing chartists, they discovered their favorite patterns lurking in the random squiggles (Malkiel 1990, pp. 135-136). Most market participants use some combination of technical and fundamental analysis, but that doesn't mean their performance is terribly successful. According to Norm Zadeh, who rates the performance of money managers, only about 1-2% of all money-slingers have a consistent and substantial record of good performance.


pages: 504 words: 139,137

Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

activist fund / activist shareholder / activist investor, algorithmic trading, Andrei Shleifer, asset allocation, backtesting, bank run, banking crisis, barriers to entry, Black-Scholes formula, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, commodity trading advisor, conceptual framework, corporate governance, credit crunch, Credit Default Swap, currency peg, David Ricardo: comparative advantage, declining real wages, discounted cash flows, diversification, diversified portfolio, Emanuel Derman, equity premium, Eugene Fama: efficient market hypothesis, fixed income, Flash crash, floating exchange rates, frictionless, frictionless market, Gordon Gekko, implied volatility, index arbitrage, index fund, interest rate swap, late capitalism, law of one price, Long Term Capital Management, margin call, market clearing, market design, market friction, merger arbitrage, money market fund, mortgage debt, Myron Scholes, New Journalism, paper trading, passive investing, price discovery process, price stability, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, random walk, Renaissance Technologies, Richard Thaler, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, selection bias, shareholder value, Sharpe ratio, short selling, sovereign wealth fund, statistical arbitrage, statistical model, stocks for the long run, stocks for the long term, survivorship bias, systematic trading, technology bubble, time value of money, total factor productivity, transaction costs, value at risk, Vanguard fund, yield curve, zero-coupon bond

To answer these questions, we first needed to know whether we were facing a liquidity spiral or an unlucky step in the random walk of an efficient market. The efficient market theory says that, going forward, prices should fluctuate randomly, whereas the liquidity spiral theory says that when prices are depressed by forced selling, prices will likely bounce back later. These theories clearly had different implications for how to position our portfolio. On Monday, we became completely convinced that we were facing a liquidity event. All market dynamics pointed clearly in the direction of liquidity and defied the random walk theory (which implies that losing every 10 minutes for several days in a row is next to impossible). Knowing that you are facing a liquidity event and that prices will eventually snap back is one thing; knowing when this will happen and what to do about it is another.

When the predictor is the dividend yield, we can also interpret the magnitude of the b coefficient. In particular, a naïve benchmark is that b = 1. This means that, if the dividend yield is one percentage point larger, then the stock return is also expected to be one percentage point larger. In other words, the dividend yield predicts the stock return because it is part of the stock return (as seen in equation 10.3), but it does not predict the price appreciation. In contrast, the random walk hypothesis b = 0 means that the price appreciation is expected to be low when the dividend yield is high, such that the overall expected equity return is independent of dividend yields. Perhaps the truth lies somewhere between these benchmarks? The data suggest otherwise. I run this regression from 1926 to 2013 with U.S. monthly data, where the monthly excess return is annualized by multiplying by 12 to make it comparable to annual dividends (the result is almost the same with 1-year forward returns, but the t-statistics must be estimated in a more complex way with overlapping data).2 The time series of the dividend yield is plotted in figure 10.1.

The losses came with remarkable consistency. Looking at the blinking screen with live P&L (profits and losses), I saw new million-dollar losses every 10 minutes for a couple of days—a clear pattern that defied the random walk theory of efficient markets and, ironically, showed remarkable likeness to my own theories. Let me explain, but let’s start from the beginning. My career as a finance guy started in 2001 when I graduated with a Ph.D. from Stanford Graduate School of Business and joined the finance faculty at the New York University Stern School of Business. My dissertation research studied how prices are determined in markets plagued by liquidity risk, and I hoped that being at a great university in the midst of things in New York City would help me find out what was going on both inside and outside the Ivory Tower.


pages: 436 words: 76

Culture and Prosperity: The Truth About Markets - Why Some Nations Are Rich but Most Remain Poor by John Kay

"Robert Solow", Albert Einstein, Asian financial crisis, Barry Marshall: ulcers, Berlin Wall, Big bang: deregulation of the City of London, business cycle, California gold rush, complexity theory, computer age, constrained optimization, corporate governance, corporate social responsibility, correlation does not imply causation, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, Donald Trump, double entry bookkeeping, double helix, Edward Lloyd's coffeehouse, equity premium, Ernest Rutherford, European colonialism, experimental economics, Exxon Valdez, failed state, financial innovation, Francis Fukuyama: the end of history, George Akerlof, George Gilder, greed is good, Gunnar Myrdal, haute couture, illegal immigration, income inequality, industrial cluster, information asymmetry, intangible asset, invention of the telephone, invention of the wheel, invisible hand, John Meriwether, John Nash: game theory, John von Neumann, Kenneth Arrow, Kevin Kelly, knowledge economy, light touch regulation, Long Term Capital Management, loss aversion, Mahatma Gandhi, market bubble, market clearing, market fundamentalism, means of production, Menlo Park, Mikhail Gorbachev, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, Naomi Klein, Nash equilibrium, new economy, oil shale / tar sands, oil shock, Pareto efficiency, Paul Samuelson, pets.com, popular electronics, price discrimination, price mechanism, prisoner's dilemma, profit maximization, purchasing power parity, QWERTY keyboard, Ralph Nader, RAND corporation, random walk, rent-seeking, Right to Buy, risk tolerance, road to serfdom, Ronald Coase, Ronald Reagan, second-price auction, shareholder value, Silicon Valley, Simon Kuznets, South Sea Bubble, Steve Jobs, telemarketer, The Chicago School, The Market for Lemons, The Nature of the Firm, the new new thing, The Predators' Ball, The Wealth of Nations by Adam Smith, Thorstein Veblen, total factor productivity, transaction costs, tulip mania, urban decay, Vilfredo Pareto, Washington Consensus, women in the workforce, yield curve, yield management

These observations are well-known, have influenced other people's assessments, and are "in the price." They are the reasons why the odds on Seabiscuit are short, the price of General Electric shares is high, mobile phone companies trade at large multiples of their current earnings, and the dollar is strong. There is powerful evidence to support the efficient market hypothesis. The theory predicts that the prices of risks will follow a "random walk." A random walk is a process in which the next step is equally likely to be in any direction. Many physical processes have these characteristics, such as the movement of particles in liquids. This is an area where models derived from statistical mechanics seem to work, and the Black-Scholes model described below is grounded in the analysis of physical systems. And numerous statistical analyses of prices in markets for securities and commodities have confirmed that they display the characteristics of a random walk.

And numerous statistical analyses of prices in markets for securities and commodities have confirmed that they display the characteristics of a random walk. In an early test of the theory, the statistician Maurice Kendall discovered that all but one of the series he studied fitted the random walk prediction. 5 It emerged that the one that did not was not in fact a series of actual market transactions but had been prepared as an average of estimated market prices. This is the kind of satisfYing confirmation of a theory that physicists often experience but is rarely available in the social sciences. The efficient market hypothesis invites a skeptical view of claims of the ability of experts to make money themselves-and even more, perhaps, of their ability to make money for other people-by trading risks. This skepticism is more readily applied to racing tipsters than to professional investment managers.

A characteristic of a market in which one side information asymmetry (buyer or seller) is better informed about the properties of the good or service than the other (seller or buyer). intellectual property Rights created by copyright, patent, or trademark legislation and associated regulations. market anomalies Observed deviations from the efficient market hypothesis. mercantilism A theory of international trade (widely held before Adam Smith and still adhered to by some devotees of DIY economics) that draws economies of scale { 364} noise trader Pareto efficiency Pareto improvement path dependency primary market productivity purchasing power parity put option random walk theory secondary market winner's curse Glossary an analogy between the exports and imports of states and the revenues and expenses of firms. A buyer or seller (especially in securities markets) whose behavior does not reflect views about the fundamental value (prospective earnings, etc.) of what he or she is buying. The property of an allocation of resources in which no one can be made better off without making someone else worse off A change that makes some people better off and no one worse off A dynamic process in which behavior is affected indefinitely by initial conditions.


pages: 612 words: 179,328

Buffett by Roger Lowenstein

asset allocation, Bretton Woods, buy and hold, cashless society, collective bargaining, computerized trading, corporate raider, credit crunch, cuban missile crisis, Eugene Fama: efficient market hypothesis, index card, index fund, interest rate derivative, invisible hand, Jeffrey Epstein, John Meriwether, Long Term Capital Management, moral hazard, Paul Samuelson, random walk, risk tolerance, Robert Shiller, Robert Shiller, Ronald Reagan, selection bias, The Predators' Ball, traveling salesman, Works Progress Administration, Yogi Berra, young professional, zero-coupon bond

In a nutshell, the theory said that at any moment, all the publicly available information about a company was reflected in the price of its stock. Whenever news about a stock became public, traders pounced, buying or selling until its price reached equilibrium. Underlying this truism was an assumption that the old price had been as “wise” as traders could make it. Therefore, the new price—and each succeeding new price—would be wise as well. The traders merely did the work of Adam Smith’s Invisible Hand. Since everything worth knowing about a company was already in the price, most security analysis was, to cite a popular text, “logically incomplete and valueless.”2 The future course of a stock would depend on new (as yet unknowable) information. A stock, then, was unpredictable; it followed a “random walk.” If markets were random, investing was a game of chance.

(For reasons known only to themselves, Merrill Lynch, Morgan Stanley, Salomon Brothers, and the rest continue to employ such soothsayers down to this day.) But the theorists replaced the chartist voodoo with a voodoo of their own. They defused the idea that prices foretold the future, but ascribed to those same prices an unerring appraisal of the present. Prices, that is, were never wrong. They incorporated, with as much perfection as humans could manage, all there was to know of a company’s long-term prospects. Studying those prospects was therefore pointless. Thus, the theorists’ attack spread from the chartists to “fundamental analysts,” such as Buffett, who combed through corporate reports looking for undervalued stocks. Quoting Fama: If the random walk theory is valid and if security exchanges are “efficient” markets, then stock prices at any point in time will represent good estimates of intrinsic or fundamental values. Thus, additional fundamental analysis is of value only when the analyst has new information … or has new insights concerning the effects of generally available information.…10 Taken at face, the qualifier offered a gaping loophole.

Bernstein, Capital Ideas: The Improbable Origins of Modem Wall Street (New York: Free Press, 1992), 115, 118–19. 4. Paul A. Samuelson, “Proof That Properly Anticipated Prices Fluctuate Randomly,” MIT Industrial Management Review, Spring 1965, pp. 782–85. 5. Paul A. Samuelson, memorandum with testimony on mutual funds, U.S. Senate, Committee on Banking and Currency, August 2, 1967. 6. Thorson, “Omahan in Search.” 7. Paul A. Samuelson. 8. Bernstein, Capital Ideas, 117. 9. Eugene F. Fama, “Random Walks in Stock Market Prices,” Financial Analysts Journal, September-October 1965. 10. Ibid. 11. “The Stock-picking Fallacy,” Economist, August 8, 1992. 12. Paul A. Samuelson, foreword to Marshall E. Blume and Jeremy J. Siegel, “The Theory of Security Pricing and Market Structure,” Journal of Financial Markets, Institutions and Instruments, Vol. 1, No. 3, 1992, pp. 1–2. 13.


pages: 464 words: 117,495

The New Trading for a Living: Psychology, Discipline, Trading Tools and Systems, Risk Control, Trade Management by Alexander Elder

additive manufacturing, Atul Gawande, backtesting, Benoit Mandelbrot, buy and hold, buy low sell high, Checklist Manifesto, computerized trading, deliberate practice, diversification, Elliott wave, endowment effect, loss aversion, mandelbrot fractal, margin call, offshore financial centre, paper trading, Ponzi scheme, price stability, psychological pricing, quantitative easing, random walk, risk tolerance, short selling, South Sea Bubble, systematic trading, The Wisdom of Crowds, transaction costs, transfer pricing, traveling salesman, tulip mania, zero-sum game

market data for profit targets in stops in Positive Directional line Positive mathematical expectation Power: of bears vs. bulls: A/D closing prices divergences Force Index MACD-Histogram MACD Line miscellaneous indicators and NH-NL zero line On-Balance Volume open interest volume profits and feeling of of trends Premiums: futures options Press, signals from Prechter, Robert Price(s). See also Closing prices; Opening prices on bar charts as consensus of value crowd behavior reflected in divergences from in Force Index indicators derived from as leader of market crowd long-term cycles in memories of of options in Random Walk theory short-term cycles in slippage support and resistance levels and understanding of volume value vs. Price risk, hedging and Price shocks Pring, Martin Private traders, see Individual traders Probabilities businessman's risk choices based on emotions vs.

People may have knowledge, but the emotional pull of the crowd often leads them to trade irrationally. A good analyst can detect repetitive patterns of crowd behavior on his charts and exploit them. Random Walk theorists claim that market prices change at random. Sure, there is a fair bit of randomness or “noise” in the markets, just as there is randomness in any crowd. Still, an intelligent observer can identify repetitive behavior patterns of a crowd and make sensible bets on their continuation or reversal. People have memories; they remember past prices, and their memories influence their decisions to buy or sell. Memories help create support under the market and resistance above it. Random Walkers deny that memories influence our behavior. As Milton Friedman pointed out, prices carry information about the availability of supply and the intensity of demand. Market participants use that information when deciding to buy or sell.

See also Japanese candlesticks Cash trades, futures compared to Catastrophic stops Ceilings, for commodities CFDs (contracts for difference) CFTC, see Commodity Futures Trading Commission Channels in A-trades Average True Range combining divergences and constructing in day-trading defined and moving averages in setting profit targets symmetrical Channel trading systems constructing channels and mass psychology standard deviation (Bollinger bands) symmetrical trading rules Chaos theory Chart analysis bar charts chaos theory detecting bias in diagonals in Efficient Market theory history of charting and insider trading Japanese candlesticks kangaroo tails “nature's law” Random Walk subjectiveness in support and resistance causes of strength of trading rules and true and false breakouts trends and trading ranges and conflicting timeframes of markets deciding to trade or wait hard right edge identifying and mass psychology as window into mass psychology Charting Commodity Market Price Behavior (L. Dee Belveal) Chart patterns: defined at right edge of charts RSI trendlines subjective interpretation of swings of mass psychology shown in Checklists Checklist Manifesto, The (Atul Gawande) Childhood, mental baggage from Churchill, Winston Classical chart analysis, see Chart analysis “Climax bottoms” Climax indicator Closing prices: Advance/Decline line on candlestick charts of daily and weekly bars of daily charts as most important consensus of value relationship of opening prices and for settlement of trading accounts Cohen, Abraham W.


Principles of Corporate Finance by Richard A. Brealey, Stewart C. Myers, Franklin Allen

3Com Palm IPO, accounting loophole / creative accounting, Airbus A320, Asian financial crisis, asset allocation, asset-backed security, banking crisis, Bernie Madoff, big-box store, Black-Scholes formula, break the buck, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, capital controls, Carmen Reinhart, carried interest, collateralized debt obligation, compound rate of return, computerized trading, conceptual framework, corporate governance, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, cross-subsidies, discounted cash flows, disintermediation, diversified portfolio, equity premium, eurozone crisis, financial innovation, financial intermediation, fixed income, frictionless, fudge factor, German hyperinflation, implied volatility, index fund, information asymmetry, intangible asset, interest rate swap, inventory management, Iridium satellite, Kenneth Rogoff, law of one price, linear programming, Livingstone, I presume, London Interbank Offered Rate, Long Term Capital Management, loss aversion, Louis Bachelier, market bubble, market friction, money market fund, moral hazard, Myron Scholes, new economy, Nick Leeson, Northern Rock, offshore financial centre, Ponzi scheme, prediction markets, price discrimination, principal–agent problem, profit maximization, purchasing power parity, QR code, quantitative trading / quantitative finance, random walk, Real Time Gross Settlement, risk tolerance, risk/return, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, Silicon Valley, Skype, Steve Jobs, The Nature of the Firm, the payments system, the rule of 72, time value of money, too big to fail, transaction costs, University of East Anglia, urban renewal, VA Linux, value at risk, Vanguard fund, yield curve, zero-coupon bond, zero-sum game, Zipcar

Three Forms of Market Efficiency You should see now why prices in competitive markets must follow a random walk. If past price changes could be used to predict future price changes, investors could make easy profits. But in competitive markets easy profits don’t last. As investors try to take advantage of the information in past prices, prices adjust immediately until the superior profits from studying past price movements disappear. As a result, all the information in past prices will be reflected in today’s stock price, not tomorrow’s. Patterns in prices will no longer exist and price changes in one period will be independent of changes in the next. In other words, the share price will follow a random walk. In competitive markets today’s stock price must already reflect the information in past prices. But why stop there? If markets are competitive, shouldn’t today’s stock price reflect all the information that is available to investors?

If markets are competitive, shouldn’t today’s stock price reflect all the information that is available to investors? If so, securities will be fairly priced and security returns will be unpredictable. No one earns consistently superior returns in such a market. Collecting more information won’t help, because all available information is already impounded in today’s stock prices. Economists define three levels of market efficiency, which are distinguished by the degree of information reflected in security prices. In the first level, prices reflect the information contained in the record of past prices. This is called weak market efficiency. If markets are efficient in the weak sense, then it is impossible to make consistently superior profits by studying past returns. Prices will follow a random walk. The second level of efficiency requires that prices reflect not just past prices but all other public information, for example, from the Internet or the financial press.

Borrowing lowers the payoff line. 12If a stock can be worth something in the future, then investors will pay something for it today, although possibly a very small amount. 13Figure 20.11 continues to assume that the exercise price on both options is equal to the current stock price. This is not a necessary assumption. Also in drawing Figure 20.11 we have assumed that the distribution of stock prices is symmetric. This also is not a necessary assumption, and we will look more carefully at the distribution of stock prices in the next chapter. 14The option values shown in Figure 20.12 were calculated by using the Black-Scholes option-valuation model. We explain this model in Chapter 21 and use it to value the Apple option. 15Here is an intuitive explanation: If the stock price follows a random walk (see Section 13-2), successive price changes are statistically independent. The cumulative price change before expiration is the sum of t random variables. The variance of a sum of independent random variables is the sum of the variances of those variables.


pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos

Andrew Wiles, Antoine Gombaud: Chevalier de Méré, beat the dealer, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, Edward Thorp, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, Johannes Kepler, lateral thinking, Myron Scholes, pattern recognition, Paul Erdős, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Rubik’s Cube, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman

If the chances of losing are greater than the chances of winning, the map of the random walk drifts downward, rather than tracking the horizontal axis. In other words, bankruptcy looms quicker. Random walks explain why gambling favours the very rich. Not only will it take much longer to go bankrupt, but there is also more chance that your random walk will occasionally meander upward. The secret of winning, for the rich or the poor, however, is knowing when to stop. Inevitably, the mathematics of random walks contains some head-popping paradoxes. In the graphs chapter 9 where Coin Man moves up or down depending on the results of a coin toss, one would expect the graph of his random walk to regularly cross the horizontal axis. The coin gives a 50:50 chance of heads or tails, so perhaps we would expect him to spend an equal amount of time either side of his starting point. In fact, though, the opposite is true.

Instead of letting Coin Man’s random walk describe a physical journey, let it represent the value of his bank account. And let the coin flip be a gamble. Heads he wins £100, tails he loses £100. The value in his bank account will swing up and down in increasingly large waves. Let us say that the only barrier that will stop Coin Man playing is when the value of his account is £0. We know it is guaranteed he will get there. In other words, he will always go bankrupt. This phenomenon – that eventual impoverishment is a certainty – is known evocatively as gambler’s ruin. Of course, no casino bets are as generous as the flipping of a coin (which has a payback percentage of 100). If the chances of losing are greater than the chances of winning, the map of the random walk drifts downward, rather than tracking the horizontal axis.

Choose a number from 0 to 7 randomly. If the number comes up, trace a line in that direction. Do this repeatedly to create a path. Venn carried this out with the most unpredictable sequence of numbers he knew: the decimal expansion of pi (excluding 8s and 9s, and starting with 1415). The result, he wrote, was ‘a very fair graphical indication of randomness’. Venn’s sketch is thought to be the first-ever diagram of a ‘random walk’. It is often called the ‘drunkard’s walk’ because it is more colourful to imagine that the original dot is instead a lamp-post and the path traced is the random staggering of a drunk. One of the most obvious questions to ask is how far will the drunk wander from the point of origin before collapsing? On average, the longer he has been walking, the further away he will be. It turns out that his distance increases with the square root of the time spent walking.


pages: 120 words: 39,637

The Little Book That Still Beats the Market by Joel Greenblatt

backtesting, index fund, intangible asset, random walk, survivorship bias, transaction costs

It is the same thing! You would be buying $10 worth of EBIT for $60, either way!43 Company A Company B Enterprise value (price + debt) 60 + 0 = $60 10 + 50 = $60 EBIT 10 10 A Random Walk Spoiled For many years, academics have debated whether it is possible to find bargain-priced stocks other than by chance. This notion, sometimes loosely referred to as the random walk or efficient market theory, suggests that for the most part, the stock market is very efficient at taking in all publicly available information and setting stock prices. That is, through the interaction of knowledgeable buyers and sellers, the market does a pretty good job of pricing stocks at “fair” value. This theory, along with the failure of most professional managers to beat the market averages over the long term,44 has understandably led to the movement toward indexing, a cost-effective strategy designed to merely match the market’s return.

The worst return during those 10+ years for following the Haugen strategy for 36 months straight (with annual turnover) was -43.1 percent. The worst 36-month period for the magic formula was +14.3 percent. Not only that, the magic formula used 69 fewer factors and a lot less math!52 So, here’s the point. The magic formula appears to perform very well. I think and hope it will continue to perform well in the future. I also hope that, just as Mark Twain aptly referred to golf as “a good walk spoiled,” perhaps someday the random walk will finally be considered spoiled as well.53 1 Bank deposits up to $100,000 are guaranteed by an agency of the U.S. government. You must hold your bank deposit or your bond until it matures (possibly 5 or 10 years, depending upon what you buy) to guarantee no loss of your original investment. 2 And yes, the dog was fine. 3 To find out what Jimbo should do, check out the box at the end of the chapter!

Each day he offers to buy your share of the business or sell you his share of the business at a particular price. Mr. Market always leaves the decision completely to you, and every day you have three choices. You can sell your shares to Mr. Market at his stated price, you can buy Mr. Market’s shares at that same price, or you can do nothing. Sometimes Mr. Market is in such a good mood that he names a price that is much higher than the true worth of the business. On those days, it would probably make sense for you to sell Mr. Market your share of the business. On other days, he is in such a poor mood that he names a very low price for the business. On those days, you might want to take advantage of Mr. Market’s crazy offer to sell you shares at such a low price and to buy Mr. Market’s share of the business. If the price named by Mr. Market is neither very high nor extraordinarily low relative to the value of the business, you might very logically choose to do nothing.


pages: 584 words: 187,436

More Money Than God: Hedge Funds and the Making of a New Elite by Sebastian Mallaby

Andrei Shleifer, Asian financial crisis, asset-backed security, automated trading system, bank run, barriers to entry, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Big bang: deregulation of the City of London, Bonfire of the Vanities, Bretton Woods, business cycle, buy and hold, capital controls, Carmen Reinhart, collapse of Lehman Brothers, collateralized debt obligation, computerized trading, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, currency manipulation / currency intervention, currency peg, Elliott wave, Eugene Fama: efficient market hypothesis, failed state, Fall of the Berlin Wall, financial deregulation, financial innovation, financial intermediation, fixed income, full employment, German hyperinflation, High speed trading, index fund, John Meriwether, Kenneth Rogoff, Kickstarter, Long Term Capital Management, margin call, market bubble, market clearing, market fundamentalism, merger arbitrage, money market fund, moral hazard, Myron Scholes, natural language processing, Network effects, new economy, Nikolai Kondratiev, pattern recognition, Paul Samuelson, pre–internet, quantitative hedge fund, quantitative trading / quantitative finance, random walk, Renaissance Technologies, Richard Thaler, risk-adjusted returns, risk/return, Robert Mercer, rolodex, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, statistical arbitrage, statistical model, survivorship bias, technology bubble, The Great Moderation, The Myth of the Rational Market, the new new thing, too big to fail, transaction costs

Of course, academia is a broad church, teaming with energetic skeptics. But from the mid 1960s to the mid 1980s, the prevailing view was that the market is efficient, prices follow a random walk, and hedge funds succeed mainly by being lucky. There is a powerful logic to this account. If it were possible to know with any confidence that the price of a particular bond or equity is likely to move up, smart investors would have pounced and it would have moved up already. Pouncing investors ensure that all relevant information is already in prices, though the next move of a stock will be determined by something unexpected. It follows that professional money managers who try to foresee price moves will generally fail in their mission. As this critique anticipates, plenty of hedge funds have no real “edge”—if you strip away the marketing hype and occasional flashes of dumb luck, there is no distinctive investment insight that allows them to beat the market consistently.

He was required to write a memo to the management explaining his miscalculations.23 The new risk-control system was connected to another rethink that followed the corn debacle: Weymar and his colleagues developed fresh respect for trends in prices. Of course, efficient-market theory holds that such trends do not exist: The random-walk consensus was so dominant that, through the 1970s and much of the 1980s, it was hard to get alternative views published in academic journals.24 But Frank Vannerson had gotten his hands on a trove of historical commodity price data that had been gathered and formatted by Dunn & Hargitt, a firm in Indiana. Before leaving Nabisco, Vannerson had spent a year working on the Dunn & Hargitt data, analyzing daily prices for fifteen commodities; and by the time Commodities Corporation opened its doors in March 1970, he had satisfied himself that price trends really did exist, no matter what academics might assert to the contrary.25 Moreover, Vannerson had devised a computer program that could trade on that finding.

Alfred Cowles, “A Revision of Previous Conclusions Regarding Stock Price Behavior,” Econometrica 28, no. 4 (October 1960). 45. By 1965, Jones’s earlier faith in charts was coming under attack even from the chartists themselves. In his 1949 essay in Fortune, Jones had singled out a Russian immigrant named Nicholas Molodovsky as “the most scientific and experimental of technical students,” reporting that with the exception of two episodes in which he had called the market wrong, “his predictions have been nearly perfect.” But in 1965 Molodovsky, by then the editor of the influential Financial Analysts Journal, commissioned a paper from a rising academic star named Eugene Fama, which appeared under the title “Random Walks in Stock Market Prices.” Fama compared chart following to astrology. By popularizing Fama’s random-walk theory, Molodovsky was burning the ground under Jones’s feet; the premise of Jones’s fund was under attack from one of its progenitors.


One Up on Wall Street by Peter Lynch

air freight, Apple's 1984 Super Bowl advert, buy and hold, corporate raider, cuban missile crisis, Donald Trump, fixed income, index fund, Irwin Jacobs, Isaac Newton, large denomination, money market fund, prediction markets, random walk, shareholder value, Silicon Valley, Y2K, Yom Kippur War, zero-sum game

Quantitative analysis taught me that the things I saw happening at Fidelity couldn’t really be happening. I also found it difficult to integrate the efficient-market hypothesis (that everything in the stock market is “known” and prices are always “rational”) with the random-walk hypothesis (that the ups and downs of the market are irrational and entirely unpredictable). Already I’d seen enough odd fluctuations to doubt the rational part, and the success of the great Fidelity fund managers was hardly unpredictable. It also was obvious that Wharton professors who believed in quantum analysis and random walk weren’t doing nearly as well as my new colleagues at Fidelity, so between theory and practice, I cast my lot with the practitioners. It’s very hard to support the popular academic theory that the market is irrational when you know somebody who just made a twentyfold profit in Kentucky Fried Chicken, and furthermore, who explained in advance why the stock was going to rise.

., 277 Noble, George, 212 Nucor, 90, 110 NutraSweet, 211 Nynex, 135 Odyssey Partners, 279 oil services industry, 151, 264 One Potato, Two, 158 OPEC, 277 options, 270–73, 280 cost of, 271 expiration of, 271–72 as insurance, 272–73 put, 273 Orion Pictures, 96 OshKosh B’Gosh, 193 over-the-counter exchange, 279 Owens Corning, 34 Pacific Telesis, 135, 213 Pampers, 107–8, 198 Pan Am, 89 Paramount, 96 Paramount Famous Lasky, 71 patents, 141 Paychex, 25, 26 PBS, 40 Pebble Beach, 40, 102, 125, 140, 209 Penn Central, 128, 134 as asset-play company, 126, 209 bankruptcy of, 122, 207 book value of, 207, 209 as turnaround company, 122, 124, 213 Pennzoil, 205 pension funds, 59, 64 pension plans, 217 People, 60 People Express, 42, 89, 269 Pep Boys, 59, 95–96, 131, 145, 190, 192, 214 Pepsi-Cola, 284 p/e ratio, see price/earnings ratio percent of sales, 198 Perot, Ross, 14, 170 Petrie, Milton, 248 Phelps Dodge, 34, 187 Philadelphia Electric, 288 Philip Morris, 129, 214, 246, 281 growth, history of, 217–18, 261 Kraft bought by, 133 negative-growth industry and, 152, 217–18 stock chart of, 262–63 Photronics, 24 Pickens, Boone, 257, 279 picks and shovels strategy, 14 Pic ’N’ Save, 59, 95, 110, 192 Piedmont Airlines, 42, 269 Pier 1 Imports, 36, 193, 247 Pizza Time Theater, 158 plastics, 119, 133 Polaroid, 49, 98n, 171–72, 254, 259, 274 portfolios: diversity of, 59–60, 239, 240 insurance for, 272–73 minimizing risk in, 241 multibaggers and, 32–33 rotating stocks in, 242–43 size of, 40, 239–41 stop orders and, 244 Postum, 71 Potter, Beatrix, 194 Premark International, 134 Prepaid Legal Services, 26 pretax profit margin, 220–21 Priam, 158 Price, Michael, 56 Price Club, 153, 268 price/earnings ratio, 165–69, 199 dot.com stocks and, 12–13 of finance companies, 200 growth rate and, 199, 218, 219 high, 165–69, 170–71 interest rates and, 172 levels of, 169, 170–71, 172 meaning of, 169 overpricing of stocks and, 168, 171–72 relativity of, 170 of stock market, 172 Primerica, 281 Pritzkers, 257 Procter and Gamble, 107–8, 109, 129, 187, 198 earnings of, 217–18 as stalwart company, 112, 115, 118 stock chart of, 115 products, demand for, 142, 254 profit margin, calculation of, 220–21 Radice, 89, 208–9 raiders, 284 see also acquisitions Ralston-Purina, 112, 118, 129, 162, 207 random-walk hypothesis, 52 Ranger Oil, 53 Raymond Industries, 212 RCA, 71, 72, 264 real estate: advantages of, 77–80 houses and, 77–80 recession of 1981–82, 86 recession of 1990, 23 Reebok, 193 Reichmanns, 256, 279 Reliance electric, 154 Remington Typewriter, 71 reports, analysts’: on Internet, 16–17 see also S&P reports reports, of companies, 194–97 see also annual reports; balance sheets Reserve Fund, 69 Resorts International, 275 restrictions, trade, 64 Retin-A, 108 Reynolds Metals, 106, 187 Rite Aid, 281 Robitussin, 142, 209 Rockefeller, John D., 66, 204 Rogers, Jimmy, 56 Rogers, Will, 54 Rukeyser, Louis, 279 Safety-Kleen, 132, 133, 137, 145, 159 sales, percent of, 198 S&P reports, 17, 21, 27, 69, 74, 123, 136, 170, 184, 197, 199, 238 Santa Fe Southern Pacific, 126, 210 Sara Lee, 37 savings accounts, 69 savings-and-loan stocks, 17, 54 savings bonds, U.S., 69 savings rates, U.S., 285 Sceilig Hotel, 28, 29 Schlumberger, 13, 34, 98, 100, 201 SCI Systems, 160 Scotty’s, 268 Scudder, Stevens and Clark, 65 Seagram, 212 Searle, 211 Sears, 59, 62, 110, 223 Securities and Exchange Commission, 64, 143 Sensormatic, 161, 223, 229–30 Service Corporation International (SCI), 35, 36, 58, 116, 137–38, 139–40 service sector, U.S., growth of, 283 7-Eleven, 42, 54, 59, 181 Seven Oaks International, 62–63, 64, 66, 131 Seven-Up, 218 shares: buybacks of, 144–45, 153, 157, 197 insider buying of, 135, 142–43 insider selling of, 143–44 see also companies; stock Shearson, 58 Shelley, Percy Bysshe, 184 Shoney’s, 116, 131, 164, 168, 261 Shop and Go, 54 shorting stocks, 273–75 Siliconix, 24 Singer, 134 Singleton, Henry E., 144 Smith, Morris, 127 SmithKline Beckman, 97–98, 99, 100, 110, 141, 187, 266 Smith Labs, 157, 158 Sorg Paper, 51 Soros, George, 56 Southland, 54 Southmark, 207 Southwestern Bell, 135 Spectrum Surveys, 136 Spielberg, Steven, 96 spinoffs, 133–36, 159 splits, 34n Sprague Tech., 134 Sprint, 135 SPUD, 158 Squibb, 281 SS Kresge, 280 SSMC, 134 Staples, 25, 26 Star Wars, 163 Steinberg, Saul, 256, 257 Sterling Drug, 108, 187 stock, indexes, 280 stock charts, 98n, 112, 164 of Avon, 166 of Bristol-Myers, 116–17 of Chrysler, 147 of Con Ed, 206 of Dow Chemical, 165 of Dreyfus, 103 of Ford Motor, 120–21 of Genesco, 156 of Home Shopping Network, 150 of Houston Industries, 113 of The Limited, 168 of Marriott, 168 of Melville, 156 of Merck, 267 of Navistar, 146 of Philip Morris, 262–63 of Procter and Gamble, 115 of Shoney’s, 168 of SmithKline Beckman, 99 of Wal-Mart Stores, 114 stock market: breaks in, 246 bullish vs. bearish, 22–23 cause and effect in, 50 distrust of, 48, 73 fluctuations of, 29–30, 82 individual stocks vs., 89–91 interest rates and, 85 Japanese, 55, 278 in 1980s, 278 of 1987–88 vs. 1929–30, 22, 282–283 in 1990s, 9, 10 in October, 1987, 29–30, 69, 86, 278, 280, 282 October, 1988 recovery of, 30 overvaluation of, 90 p/e ratio of, 172 predictability of, 84–88 preparedness for, 86–87 random-walk hypothesis of, 52 as stud poker game, 74–75, 76 theories of, 52 trading hours of, 278 turnover in, 281 volume of, 281 weak, 33 world events and, 276–80, 283 see also investment; stocks stocks: annual gain of, 72, 85 approved lists of, 59–60 average return of, 237–38 bargain, 261–64 blue-chip, see blue-chip stocks bonds vs., 69–70, 88, 112, 237 cash flow and, 214 charts of, see stock charts choosing, 95–97, 98, 231–33; see also investment classification of, 110–27 comebacks of, 264 common misconceptions of, 258–69 conservative, 265; see also blue-chip stocks diluting of, 145 dividends and, see dividends dot.com, see dot.com stocks efficient market hypothesis of, 52 falling of, 259–60, 269–70 fluctuations of, 29–30, 82 frequent trading of, 238–39 hot, 149–52 initial public offering (IPO) of, 159 insider buying of, 135, 142–43 insider selling of, 143–44, 180 institutional ownership of, 55, 57, 136, 142–45, 179 Internet and, 10–12 length of ownership of, 112, 115, 266 market vs., 89–91 money-market funds vs., 69, 88 overpricing of, and p/e ratio, 168, 171–72 portfolios of, see portfolios public attitude toward, 47–48, 73 real estate vs., 78–80 rising of, 260–61, 269–70 risk of, 71–76, 80 shorting of, 273–75 summary evaluation of, 227–33 whisper, 157–59, 280 see also companies; investment; stock market stock tables, 165–68 Stop & Shop, 59, 128, 163, 261 stock, return of, 33 stop orders, 244 Storer Communications, 101–2, 256 street lag, 57–60, 101–2 Student Loan Marketing, 247 Subaru, 33, 34n, 36, 58, 95, 115, 251, 260 Sullivan, D.

He wouldn’t have won any awards from Ms. magazine. I was thrilled to be hired at Fidelity, and also to be installed in Gerry Tsai’s old office, after Tsai had departed for the Manhattan Fund in New York. Of course the Dow Jones industrials, at 925 when I reported for work the first week of May, 1966, had fallen below 800 by the time I headed off to graduate school in September, just as the Lynch Law would have predicted. RANDOM WALK AND MAINE SUGAR Summer interns such as me, with no experience in corporate finance or accounting, were put to work researching companies and writing reports, the same as the regular analysts. The whole intimidating business was suddenly demystified—even liberal arts majors could analyze a stock. I was assigned to the paper and publishing industry and set out across the country to visit companies such as Sorg Paper and International Textbook.


pages: 500 words: 145,005

Misbehaving: The Making of Behavioral Economics by Richard H. Thaler

"Robert Solow", 3Com Palm IPO, Albert Einstein, Alvin Roth, Amazon Mechanical Turk, Andrei Shleifer, Apple's 1984 Super Bowl advert, Atul Gawande, Berlin Wall, Bernie Madoff, Black-Scholes formula, business cycle, capital asset pricing model, Cass Sunstein, Checklist Manifesto, choice architecture, clean water, cognitive dissonance, conceptual framework, constrained optimization, Daniel Kahneman / Amos Tversky, delayed gratification, diversification, diversified portfolio, Edward Glaeser, endowment effect, equity premium, Eugene Fama: efficient market hypothesis, experimental economics, Fall of the Berlin Wall, George Akerlof, hindsight bias, Home mortgage interest deduction, impulse control, index fund, information asymmetry, invisible hand, Jean Tirole, John Nash: game theory, John von Neumann, Kenneth Arrow, Kickstarter, late fees, law of one price, libertarian paternalism, Long Term Capital Management, loss aversion, market clearing, Mason jar, mental accounting, meta analysis, meta-analysis, money market fund, More Guns, Less Crime, mortgage debt, Myron Scholes, Nash equilibrium, Nate Silver, New Journalism, nudge unit, Paul Samuelson, payday loans, Ponzi scheme, presumed consent, pre–internet, principal–agent problem, prisoner's dilemma, profit maximization, random walk, randomized controlled trial, Richard Thaler, Robert Shiller, Robert Shiller, Ronald Coase, Silicon Valley, South Sea Bubble, Stanford marshmallow experiment, statistical model, Steve Jobs, Supply of New York City Cabdrivers, technology bubble, The Chicago School, The Myth of the Rational Market, The Signal and the Noise by Nate Silver, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions, transaction costs, ultimatum game, Vilfredo Pareto, Walter Mischel, zero-sum game

., 29 surveys used in experiments of, 38 psychological accounting, see mental accounting “Psychology and Economics Conference Handbook,” 163 “Psychology and Savings Policies” (Thaler), 310–13 Ptolemaic astronomy, 169–70 public goods, 144–45 Public Goods Game, 144–46 Punishment Game, 141–43, 146 Pythagorean theorem, 25–27 qualified default investment alternatives, 316 quantitative analysis, 293 Quarterly Journal of Economics, 197, 201 quasi-hyperbolic discounting, 91–92 quilt, 57, 59, 61, 65 Rabin, Matthew, 110, 181–83, 353 paternalism and, 323 racetracks, 80–81, 174–75 Radiolab, 305 randomized control trials (RCTs), 8, 338–43, 344, 371 in education, 353–54 Random Walk Down Walk Street, A (Malkiel), 242 rational expectations, 98, 191 in macroeconomics, 209 rational forecasts, 230–31 rationality: bounded, 23–24, 29, 162 Chicago debate on, 159–63, 167–68, 169, 170, 205 READY4K!, 343 real business cycle, 191 real estate speculation, 372 rebates, 121–22, 363 recessions, 131–32 fiscal policy in, 209 reciprocity, 182 Reder, Mel, 159 Reeves, Richard, 330, 332 reference price, 59, 61–62 regression toward the mean, 222–23 research and development, 189 reservation price, 150 retirement, savings for, see savings, for retirement return, risk vs., 225–29 returns, discounts and, 242 revealed preferences, 86 “right to carry” law, 265n risk: measurement of, 225–29 return vs., 225–29 “Risk and Uncertainty: A Fallacy of Large Numbers” (Samuelson), 194 risk aversion, 28–29, 33, 83, 84 crowds and, 301, 369 on Deal or No Deal, 298–99 equity premium and, 191–92 of managers, 190–91 moderate vs. extreme, 298–99 risk premium, 14–16, 226 irrationality of, 16–17 risk-seeking behavior, 81, 83 roadside stands, 146–47 Robie House, 270 Rochester, University of, 41, 51, 205, 216 Roger and Me (film), 122 rogue traders, 84 Roll, Richard, 167, 208 Romer, David, 292 Rosen, Sherwin, 12, 15, 17, 21, 35, 42, 321 at behavioral economics debate, 159 Rosett, Richard, 17, 34, 46, 68, 73 Ross, Lee, 181 Ross, Steve, 167 Roth, Alvin, 130, 148 Royal Dutch Shell, 248, 249, 251 rules (in self-control), 106–9, 111 Russell, Thomas, 18, 203 Russell Sage Foundation, 177–78, 179, 181, 185 Russell Sage summer camps, 181–84, 199 Russian roulette, 13–14 Russo, Jay, 122 S&P 500, 232, 233 Sadoff, Sally, 354 safety, paying for, 13–14 St.

Look around”: Quoted in Fox (2009), p. 199. 240 more rigorous, thorough, and polite version of the “idiots” paper: De Long et al. (1990). 241 “an expensive monument”: Graham ([1949] 1973), p. 242. 242 That is exactly what we found: Lee, Shleifer, and Thaler (1991). 242 thesis on closed-end funds: Thompson (1978). 242 A Random Walk Down Wall Street: Malkiel (1973). 243 “But they can’t”: Chen, Kan, and Miller (1993), p. 795. 243 the last set of stones: The five papers are: Lee, Shleifer, and Thaler (1991), Chen, Kan, and Miller (1993a), Chopra et al. (1993a), Chen, Kan, and Miller (1993b), and Chopra et al. (1993b). Chapter 26: Fruit Flies, Icebergs, and Negative Stock Prices 249 LTCM had collapsed: Lowenstein (2000). 249 in a paper they published on this topic: Shleifer and Vishny (1997). 250 an academic paper about the . . . episode: Lamont and Thaler (2003). 251 “we might define an efficient market”: Black (1986), p. 553. 252 “liar loans”: See Mian and Sufi (2014).

., Paul Andreassen, and Stanley Schachter. 1985. “II. Random and Non-Random Walks on the New York Stock Exchange.” Journal of Economic Behavior and Organization 6, no. 4: 331–8. Hsee, Christopher K., Yang Yang, Yangjie Gu, and Jie Chen. 2009. “Specification Seeking: How Product Specifications Influence Consumer Preference.” Journal of Consumer Research 35, no. 6: 952–66. Internal Revenue Service. 1998. “Revenue Ruling 98–30.” Internal Revenue Bulletin 25 (June 22): 8–9. Available at: http://www.irs.gov/pub/irs-irbs/irb98-25.pdf. Jackson, Eric. 2014. “The Case For Apple, Facebook, Microsoft Or Google Buying Yahoo Now.” Forbes.com, July 21. Available at: http://www.forbes .com/sites/ericjackson/2014/07/21/the-case-for-apple-facebook-micro soft-or-google-buying-yahoo-now. Jensen, Michael C. 1969. “Risk, The Pricing of Capital Assets, and the Evaluation of Investment Portfolios.”


pages: 348 words: 83,490

More Than You Know: Finding Financial Wisdom in Unconventional Places (Updated and Expanded) by Michael J. Mauboussin

Albert Einstein, Andrei Shleifer, Atul Gawande, availability heuristic, beat the dealer, Benoit Mandelbrot, Black Swan, Brownian motion, butter production in bangladesh, buy and hold, capital asset pricing model, Clayton Christensen, clockwork universe, complexity theory, corporate governance, creative destruction, Daniel Kahneman / Amos Tversky, deliberate practice, demographic transition, discounted cash flows, disruptive innovation, diversification, diversified portfolio, dogs of the Dow, Drosophila, Edward Thorp, en.wikipedia.org, equity premium, Eugene Fama: efficient market hypothesis, fixed income, framing effect, functional fixedness, hindsight bias, hiring and firing, Howard Rheingold, index fund, information asymmetry, intangible asset, invisible hand, Isaac Newton, Jeff Bezos, Kenneth Arrow, Laplace demon, Long Term Capital Management, loss aversion, mandelbrot fractal, margin call, market bubble, Menlo Park, mental accounting, Milgram experiment, Murray Gell-Mann, Nash equilibrium, new economy, Paul Samuelson, Pierre-Simon Laplace, quantitative trading / quantitative finance, random walk, Richard Florida, Richard Thaler, Robert Shiller, Robert Shiller, shareholder value, statistical model, Steven Pinker, stocks for the long run, survivorship bias, The Wisdom of Crowds, transaction costs, traveling salesman, value at risk, wealth creators, women in the workforce, zero-sum game

The bad news, as physicist Phil Anderson notes above, is that the tails of the distribution often control the world. Tell Tail Normal distributions are the bedrock of finance, including the random walk, capital asset pricing, value-at-risk (VaR), and Black-Scholes models. Value-at-risk models, for example, attempt to quantify how much loss a portfolio may suffer with a given probability. While there are various forms of VaR models, a basic version relies on standard deviation as a measure of risk. Given a normal distribution, it is relatively straightforward to measure standard deviation, and hence risk. However, if price changes are not normally distributed, standard deviation can be a very misleading proxy for risk.2 The research, some done as far back as the early 1960s, shows that price changes do not follow a normal distribution. Exhibit 31.1 shows the frequency distribution of S&P 500 daily returns from January 1, 1978, to March 30, 2007, and a normal distribution derived from the data.

Inversely, the less frequently investors evaluate their portfolios, the more likely they are to see gains. Exhibit 8.1 provides some numbers to illustrate these concepts.6 The basis for this analysis is an annual geometric mean return of 10 percent and a standard deviation of 20.5 percent (nearly identical to the actual mean and standard deviation from 1926 through 2006).7 The table also assumes that stock prices follow a random walk (an imperfect but workable assumption) and a loss-aversion factor of 2. (Utility = Probability of a price increase - probability of a decline x 2.) EXHIBIT 8.1 Time, Returns, and Utility Source: Author analysis. A glance at the exhibit shows that the probability of a gain or a loss in the very short term is close to fifty/fifty. Further, positive utility—essentially the avoidance of loss aversion—requires a holding period of nearly one year.

As a teenager in the Pacific Coast League, DiMaggio had a sixty-one-game streak. Of note, too, is immediately after DiMaggio’s fifty-six-game streak was broken, he went on to a sixteen-game hitting streak. So he got a hit in seventy-two of seventy-three games during the course of the 1941 season. 8 Here’s a sample of some references (there are too many to list exhaustively): Burton G. Malkiel, A Random Walk Down Wall Street (New York: W. W. Norton & Company, 2003), 191; Nassim Taleb, Fooled By Randomness: The Hidden Role of Chance in Markets and in Life (New York: Texere, 2001), 128-131; Gregory Baer and Gary Gensler, The Great Mutual Fund Trap (New York: Broadway Books, 2002), 16-17; Peter L. Bernstein, Capital Ideas: The Improbable Origins of Modern Wall Street (New York: Free Press, 1992), 141-43. 9 Baer and Gensler, The Great Mutual Fund Trap, 17.


pages: 261 words: 86,905

How to Speak Money: What the Money People Say--And What It Really Means by John Lanchester

asset allocation, Basel III, Bernie Madoff, Big bang: deregulation of the City of London, bitcoin, Black Swan, blood diamonds, Bretton Woods, BRICs, business cycle, Capital in the Twenty-First Century by Thomas Piketty, Celtic Tiger, central bank independence, collapse of Lehman Brothers, collective bargaining, commoditize, creative destruction, credit crunch, Credit Default Swap, crony capitalism, Dava Sobel, David Graeber, disintermediation, double entry bookkeeping, en.wikipedia.org, estate planning, financial innovation, Flash crash, forward guidance, Gini coefficient, global reserve currency, high net worth, High speed trading, hindsight bias, income inequality, inflation targeting, interest rate swap, Isaac Newton, Jaron Lanier, joint-stock company, joint-stock limited liability company, Kodak vs Instagram, liquidity trap, London Interbank Offered Rate, London Whale, loss aversion, margin call, McJob, means of production, microcredit, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, negative equity, neoliberal agenda, New Urbanism, Nick Leeson, Nikolai Kondratiev, Nixon shock, Northern Rock, offshore financial centre, oil shock, open economy, paradox of thrift, plutocrats, Plutocrats, Ponzi scheme, purchasing power parity, pushing on a string, quantitative easing, random walk, rent-seeking, reserve currency, Richard Feynman, Right to Buy, road to serfdom, Ronald Reagan, Satoshi Nakamoto, security theater, shareholder value, Silicon Valley, six sigma, Social Responsibility of Business Is to Increase Its Profits, South Sea Bubble, sovereign wealth fund, Steve Jobs, survivorship bias, The Chicago School, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, trickle-down economics, Washington Consensus, wealth creators, working poor, yield curve

If you look at the companies’ respective earnings, a share of Apple costs just over 11 times what the company earned last year, whereas a share of Amazon is valued at 3,500 times earnings. In other words, Amazon is 300 times more expensive than Apple. That might seem nuts, but the price is based on the idea that in the future, Amazon will earn huge amounts of money, so you buy the share now in order to get in early for the huge takeoff that is going to come. Apple on the other hand is more of a known quantity, so you are getting what you pay for. It’s very difficult to know what the realistic P/E ratio is for any stock: as Burton Malkiel put it in his efficient-market theory investment classic, A Random Walk Down Wall Street, “God Almighty does not know the proper price-earnings-multiple for a common stock.” 63 Historically, companies with low P/E ratios—what are known as “value stocks”—have tended to outperform those with high P/Es, in part because a high P/E implies high expectations that are easily disappointed.

Commenting on her in 1979, he said, “This election was about a woman who believes in inequality, passionately, who isn’t Keynesian, who is not worried about dole queues.” In his biography of Thatcher, Charles Moore says that in Walden’s view, “if interviewers had wanted to find the truth, they should have asked her, ‘Mrs Thatcher, do you believe in a more unequal society?’ ” # Book recommendation: Burton Malkiel’s classic A Random Walk Down Wall Street lays out a thoroughly convincing explication of the thesis, with lots of practical advice for private investors. Part II A LEXICON OF MONEY the aaaaa number A term I’ve just made up to denote 16,438, for the purpose of making sure it comes first in this lexicon. This number is, in the words of Melinda Gates, “the most important statistic in the world.” It’s the number of children under five who aren’t dying every day, compared with the number who were dying daily in 1990.

Alice Shroeder’s The Snowball, a biography of Warren Buffett, is very different in tone and texture, but it brings in a lot of stories and information from the world of finance, as does Sebastian Mallaby’s More Money Than God, a (suprisingly and convincingly positive) study of hedge funds. Some of you may well be thinking: but how is any of this going to help me become rich? If you are, here are two books for you: Burton Malkiel’s A Random Walk Down Wall Street, which explains efficient-market theory for the ordinary investor, and John Kay’s The Long and Short of It. Kay’s book is the best book ever written for the British individual investor, by a country mile. Ben Graham’s The Intelligent Investor, the first book written on the subject, remains one of the best. One of the liveliest areas of argument in this field concerns the poorest people in the world, and the question of how best to help them.


pages: 407 words: 114,478

The Four Pillars of Investing: Lessons for Building a Winning Portfolio by William J. Bernstein

asset allocation, Bretton Woods, British Empire, business cycle, butter production in bangladesh, buy and hold, buy low sell high, carried interest, corporate governance, cuban missile crisis, Daniel Kahneman / Amos Tversky, Dava Sobel, diversification, diversified portfolio, Edmond Halley, equity premium, estate planning, Eugene Fama: efficient market hypothesis, financial independence, financial innovation, fixed income, George Santayana, German hyperinflation, high net worth, hindsight bias, Hyman Minsky, index fund, invention of the telegraph, Isaac Newton, John Harrison: Longitude, Long Term Capital Management, loss aversion, market bubble, mental accounting, money market fund, mortgage debt, new economy, pattern recognition, Paul Samuelson, quantitative easing, railway mania, random walk, Richard Thaler, risk tolerance, risk/return, Robert Shiller, Robert Shiller, South Sea Bubble, stocks for the long run, stocks for the long term, survivorship bias, The inhabitant of London could order by telephone, sipping his morning tea in bed, the various products of the whole earth, the rule of 72, transaction costs, Vanguard fund, yield curve, zero-sum game

., 100 Precious metals stocks, 123–124, 155 Present value vs. discount rate, discounted dividend model (DDM), 46–48 Press coverage, 219–225 Prestiti, Venetian, 10–13 Price, annuity, 9–13 Price-to-earnings (P/E) ratio, 58, 68–69, 150, 174-175 Prices, stock (See Stock prices) Primerica, 83 Principal transaction, 196 Principia Pro software, Morningstar Inc., 98, 152, 205 Prudential-Bache, 200 Psychology of investing (Pillar 3) (See Behavioral economics) Purchase vs. investment, 45 Quinn, Jane Bryant, 220, 221 Radio Corporation of America, 132, 147 Railroad bubble, 143-145, 158, 159–160 “Railway time,” 144 “Random walk,” 25 A Random Walk Down Wall Street (Malkiel), 224 Randomness in market, 25, 175–177, 186 (See also Performance) Raskob, John J., 65, 147, 148 RCA, 132, 147 Real Estate Investment Trusts (REITs), 69, 72, 109, 123, 124, 250, 254, 263, 296 Real (inflation-adjusted) returns bonds, twentieth century, 19 discounted dividend model (DDM) for different instruments, 68–69 establishment of, 7 future outlook, 67–71 retirement investments, 230 retirement withdrawal strategies, 231–234 stock, 26 and young savers, 238–239 Realized returns, 71–73 Rebalancing, 286-292 Regan, Donald, 194 Regret avoidance, 177 Reinvesting income (benefits of), 61 REITs (Real Estate Investment Trusts), 69, 72, 109, 123, 124, 250, 254, 263, 296 Retained earnings and dividends paid, 59–60 Retirement planning, 229–241 end-period wealth, 26–27 immortality assumption, 229–235 impact of crash in stock market, 61-62 portfolio rebalancing, 276, 282, 285, 286-293 vs. young savers, 236–239 Returns in brokerage accounts, 198–199, 200 calculation of, 186–187n1 expected (See Expected returns) and market capitalization, 32–34 mutual funds, 203-208 rebalanced, 286-293 Risk bond prices, 11-20 company quality, 34–38 cyclical companies, 64 defined, 11 discounted dividend model (DDM), 41-42 historic record as gauge of, 32 interest rates, 13, 260 long-term, 22-29 and market capitalization, 34 and measurement, 22–29 Risk-return relationship diversification and rebalancing, 286-291 historical perspective, 6–13, 22-29, 38 retirement years, 231–236 short- vs. long-term risk and behavioral economics, 172–173, 184-185 summary, by investment type, 38–39 Risk premium, 184 Riskless assets, 110, 114, 260, 264 Rockefeller, Percy, 147 Rocket (Stephenson), 143 Roman Empire, interest rates in, 8–9 Russell 2000, 248 Russell 3000, 245, 246 Safety penalty, 184 Sales training for brokers, 200 Samuelson, Paul, 214 Sanborn, Robert, 84–85 Santayana, George, 6, 129 Sarnoff, Mrs.

If you do, then you’re also likely quite adept at seeing the George Washington Bridge or the face of Bruce Willis in the clouds scudding overhead. The pattern of annual stock returns is almost totally random and unpredictable. The return in the last year, or the past five years, gives you no hint of next year’s return—it is a “random walk.” As we’ll see later, no one—not the pundits from the big brokerage firms, not the newsletter writers, not the mutual fund managers, and certainly not your broker—can predict where the market will go tomorrow or next year. So the twentieth century has seen three severe drops in stock prices, one of them catastrophic. The message to the average investor is brutally clear: expect at least one, and perhaps two, very severe bear markets during your investing career. Long-term risk—the probability of running out of money over the decades—is an entirely different matter.

Bary, Andrew, “Vertigo: The New Math Behind Internet Capital’s Stock Price is Fearsome.” Barrons, January 10, 2000. Brooks, John, Once in Golconda. Wiley, 1999. Chamberlain, Lawrence, and Hay, William W., Investment and Speculation. New York, 1931. Chancellor, Edward, Devil Take the Hindmost. Penguin, 1999. Galbraith, John K., The Great Crash. Houghton Mifflin, 1988. Johnson, Paul M., The Birth of the Modern: World Society 1815–1830. Harper Collins, 1991. Kindleberger, Charles P., Manias, Panics, and Crashes. Wiley, 2000. Mackay, Charles, Extraordinary Popular Delusions and the Madness of Crowds. Harmony Books, 1980. Maddison, Angus, Monitoring the World Economy 1820-1992. OECD, 1995. Malkiel, Burton G., A Random Walk Down Wall Street. W. W. Norton, 1996. Nocera, Joseph, A Piece of the Action.


pages: 440 words: 108,137

The Meritocracy Myth by Stephen J. McNamee

affirmative action, Affordable Care Act / Obamacare, American ideology, Bernie Madoff, British Empire, business cycle, collective bargaining, computer age, conceptual framework, corporate governance, deindustrialization, delayed gratification, demographic transition, desegregation, deskilling, equal pay for equal work, estate planning, failed state, fixed income, gender pay gap, Gini coefficient, glass ceiling, helicopter parent, income inequality, informal economy, invisible hand, job automation, joint-stock company, labor-force participation, longitudinal study, low-wage service sector, marginal employment, Mark Zuckerberg, mortgage debt, mortgage tax deduction, new economy, New Urbanism, obamacare, occupational segregation, old-boy network, pink-collar, plutocrats, Plutocrats, Ponzi scheme, post-industrial society, prediction markets, profit motive, race to the bottom, random walk, school choice, Scientific racism, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, The Spirit Level, The Wealth of Nations by Adam Smith, too big to fail, trickle-down economics, upwardly mobile, We are the 99%, white flight, young professional

The best models account for only about half of the variance in income attainment and about two-thirds of the variance in occupational attainment. Some of the variance “unexplained” by these models could come from a combination of leaving out factors that matter and from less-than-perfect measures of the factors included. But some of the unexplained or residual variation is also likely due to simple random variation—or, in more everyday language, “luck.” The Random-Walk Hypothesis So far in this chapter, our discussion has revolved around education, jobs, and income. We have argued that the “going rate” of return for the jobs that people hold depends, at least in part, on factors that lie outside the control of individual workers themselves. Getting ahead in terms of the occupations people hold and the pay they receive involves an element of luck—being in the right place at the right time.

., cruise lines, medical and assisted-living services). Finally, developmental disequilibrium refers to unequal conditions of development in different countries that create opportunities to introduce products and services available in one place that are not yet available in another place. To some extent, those who are the most clever or most insightful might be better able to anticipate various market shakeups. However, the “random-walk hypothesis” developed by economists seems to account best for who ends up with the right idea, the right product, or the right service. The argument is simply that striking it rich tends to be like getting struck by lightning: many are walking around, but only a few get randomly struck. Large fortunes tend to be made quickly, taking early advantage of market shakeups. The window for striking it rich is very narrow since once it is open, others quickly rush in.

Clearly, some individual entrepreneurs who strike out on their own manage to defy the odds and do very well. Those who succeed do not necessarily work harder than those who fail; nor are they necessarily more inherently capable or meritorious. Having sufficient start-up capital to launch new enterprises (it takes money to make money) and being in the right place at the right time with the right idea (random-walk hypothesis) do, however, have a great deal to do with entrepreneurial success. The Case of Microsoft In rare circumstances, such individuals may take advantage of temporary market imbalances and launch new enterprises that start out small but evolve into corporate giants. One particularly prominent example is the establishment of the Microsoft Corporation in 1975, now the forty-second largest corporation in the world (Forbes 2012a).


Concentrated Investing by Allen C. Benello

activist fund / activist shareholder / activist investor, asset allocation, barriers to entry, beat the dealer, Benoit Mandelbrot, Bob Noyce, business cycle, buy and hold, carried interest, Claude Shannon: information theory, corporate governance, corporate raider, delta neutral, discounted cash flows, diversification, diversified portfolio, Edward Thorp, family office, fixed income, high net worth, index fund, John von Neumann, Louis Bachelier, margin call, merger arbitrage, Paul Samuelson, performance metric, random walk, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, survivorship bias, technology bubble, transaction costs, zero-sum game

Like Shannon, he too sought to exploit the random walk phenomenon, not through Shannon’s Demon, but through convertible arbitrage. Thorp’s strategy, though it was considerably more sophisticated, shared with Shannon’s Demon the use of constant rebalancing to eke out tiny profits from changes in the prices of matched securities as they reverted to the mean. In his 1967 book Beat The Market, his follow-up to Beat the Dealer, Thorp also described his investment process as a “a scientific stock market system” perhaps as a nod to Shannon’s scientific investing lectures. The difference was that Thorp’s strategy sought arbitrage opportunities between baskets of securities that by implication should have traded at the same price. For example, in 1974, American Motors convertible bonds were selling for the same price as the stock underlying them and paying 8.3 percent.

In the mid‐1960s, he started giving talks at MIT on the subject of scientific investing.22 Scientific investing did not mean technical analysis. Shannon had tinkered with technical charting in the early 1960s, but had rejected it, describing the price charts used by technicians as “a very noisy reproduction of the important data.”23 Rather, Shannon lectured on statistical methods for profiting from a stock’s random walk. One such method was what Poundstone named Shannon’s Demon. The idea was to form a portfolio of equal parts cash and a stock, and rebalance regularly to take advantage of the stock’s randomly jittering price movements. Shannon’s Demon worked as follows: Let’s say we begin with a portfolio of $10,000: $5,000 will be held in cash, and $5,000 will be invested in a stock at noon. At noon the next day, the portfolio is rebalanced.

., 143 Norwegian Cruise Line, 146–151 Noyce, Robert, 161–162 O Oak Value Capital Management, 18 Odegard, Jan Tore, 132–133 Oian, Anne, 147 Olsen, Fred, 132, 135, 136–137 Olsen Group, 132, 136 opposed risks, Keynes on, 37, 43 O’Shaughnessy, Patrick, 103–104, 211 230 P permanent capital risk and permanent impairment of capital, 63–64 Siem on, 154–155, 155–156 temperament of investors and, 2, 208–209, 211–214 Peters, Betty, 117 Phillips Petroleum, 138, 139 Pidgeon, Larry, 166 Pigou, Arthur, 40 Portfolio Problem, The (Shannon), 74 positive free cash flow, 18–19 Poundstone, William, 72–73, 77, 78, 80–81 P.R. Finance Company, 44–45 price-to-book value, 101 price to earnings ratio earnings yield compared to, 49 Simpson on, 19 Princeton-Newport Partners, 83 Provincial Insurance Company, 38, 60, 206, 212 R random walk convertible arbitrage, 83–87 Shannon’s Demon, 79–83, 80 rebalancing, of uncorrelated assets, 80 Reebok, 23 Ringdal, 134–135 RJ Reynolds, 163 Roberts, Brian, 177 Robertson, Julian, 144 Index Rosenfield, Joe, 8–9, 159–169, 172 Ross, Arthur, 184–186, 190 Ruane, William J., 165 S Saks Fifth Avenue, 180, 181 Salomon Brothers, 25 Samuelson, Paul, 84–85 Sanborn Map, 89–93, 105 Schloss, Walter, 120–121 scientific investing, 79 Scott, Francis C., 39, 45, 49, 60, 206, 212 scuttlebutt method, 116, 215 Securities and Exchange Commission (SEC), 117–118 Security Analysis (Graham), 37, 38, 45 See, Harry, 114–115 See’s Candies, 113–118, 126, 127, 195, 214–215 Selfridge, John, 72 Sequoia Fund, 162, 165, 168, 172 Shannon, Claude, 73–74, 77, 78–83 Shannon’s Demon, 79–83, 80 Shapiro, John, 177–178, 185, 189 Shareholders Management, 10–11 Shaw Communications, 194–196 Shiller, Robert J., 38 shorting, Keynes on, 41 Siem, Ivan, 136 Siem, Kristian, 131–158 DSND Subsea and, 151–153 offshore drilling by, 131–138, 143–146 shipping and cruise lines, 138– 142, 146–151 231 Index temperament for investing by, 205–206, 213, 214, 215–216 on valuation, 153–155, 155–156 Siem Industries, 145–151, 155–156 Simpson, Lou, 5–33 “conservative, concentrated” investment approach of, 25–29, 194 early biographical information, 9–11 GEICO investment decisions by, 11–15, 18–29 GEICO results of, 15–18, 17 Grinnell and, 166 hired as GEICO chief investment officer, 5–9 investment philosophy of, 18–25 overview of portfolio concentration used by, 1, 4 temperament for investing by, 123–124, 204–205, 213, 214, 215–216 Singleton, Henry, 78 $64,000 Question, The (television show), 74, 76–77 Smith, E.


pages: 483 words: 141,836

Red-Blooded Risk: The Secret History of Wall Street by Aaron Brown, Eric Kim

activist fund / activist shareholder / activist investor, Albert Einstein, algorithmic trading, Asian financial crisis, Atul Gawande, backtesting, Basel III, Bayesian statistics, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Black Swan, business cycle, capital asset pricing model, central bank independence, Checklist Manifesto, corporate governance, creative destruction, credit crunch, Credit Default Swap, disintermediation, distributed generation, diversification, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, experimental subject, financial innovation, illegal immigration, implied volatility, index fund, Long Term Capital Management, loss aversion, margin call, market clearing, market fundamentalism, market microstructure, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, natural language processing, open economy, Pierre-Simon Laplace, pre–internet, quantitative trading / quantitative finance, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, road to serfdom, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, special drawing rights, statistical arbitrage, stochastic volatility, stocks for the long run, The Myth of the Rational Market, Thomas Bayes, too big to fail, transaction costs, value at risk, yield curve

Its only risks now are that there might be some problem with the futures clearinghouse or some mismatch between the Treasuries it holds and the Treasuries deliverable under the futures contracts. Otherwise, it does not care if Treasury prices go up or down, or even if the U.S. government defaults. We always knew there were some risks to this kind of leverage, but they seemed much smaller than the risks you eliminated by hedging. We learned that was not necessarily true. In a severe credit crunch and liquidity crisis, even good leverage, the kind that offsets your risks, could kill. The next step is to think like a frequentist. What things did other people learn that were really just fluctuations in a random walk? U.S. Treasury bonds did great during the crisis, but that might not happen next time. A lot of people decided that illiquid investments were bad, without distinguishing carefully between the disaster of investments that were supposed to be liquid but weren’t versus investments everyone knew were illiquid all along.

To predict the future, you string together randomly selected days from the past—with replacement, meaning you can pick the same past day more than once. You do this many times to generate a distribution of possible future outcomes. Simple resampling works only when the data are independent—that is, when yesterday’s move doesn’t tell you anything about today’s. Another name for a series with independent changes is a random walk, which of course is one of the famous models in finance. I believed, however, that financial time series were typically not random walks. One kind of common deviation from a random walk is called autocorrelation. That means yesterday’s move tells you something about today’s move—up days are followed by other up days either more (positive autocorrelation) or less (negative autocorrelation) than half the time. Both kinds of autocorrelation are observed in financial time series, but back in 1980 I believed they had to be minor.

Suppose I study all the people making markets (that is, setting prices at which they will buy from or sell to anyone) in, say, oil futures. They set different prices, implying different betting odds on oil prices in the future. I’m only interested in the market makers generating consistent profits. But even within this group there is a variety of prices and also differences in the positions they have built up. You might argue that the differences in prices are going to be pretty small, and some kind of average or market-clearing price is the best estimate of probability. One issue with this is none of the prices directly measure probability; all of them blend in utility to some degree. A person who locks in a price for future oil may not believe the price of oil is going up. She may be unable to afford higher prices if that does happen, and is willing to take an expected loss in order to ensure survival of her business.


pages: 447 words: 104,258

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

algorithmic trading, asset allocation, asset-backed security, backtesting, banking crisis, Black Swan, Black-Scholes formula, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, discrete time, diversification, fixed income, implied volatility, interest rate derivative, interest rate swap, margin call, market microstructure, martingale, p-value, passive investing, quantitative trading / quantitative finance, random walk, risk/return, Satyajit Das, Sharpe ratio, short selling, statistical model, stochastic process, stochastic volatility, time value of money, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-coupon bond

MGARCH see multivariate GARCH process mixed data sampling (MIDAS) process mixed jump diffusion model model risk modified duration (MD) modified VaR (MVaR) moments CAPM money markets moneyness Monte Carlo simulation accuracy exotic options fractional Brownian motion jump processes sensitivities simulation examples VaR Moody’s rating agency mortgage-backed securities Morton see Heath, Jarrow and Morton model moving average (MA) process moving averages ARIMA process ARMA process MA process MSCI Barra MtM see marked to market multivariate GARCH (MGARCH) process MVaR see modified VaR NASDAQ index NDFs see non-deliverable forwards NDOs see non-delivery options neural networks (NNs) “no arbitrage” condition non-deliverable forwards (NDFs) non-delivery options (NDOs) non-financial commodity futures non-linear models non-path dependent options non-stationary processes normal distribution Norwegian krone (NOK) OECD see Organisation for Economic Co-operation and Development offer price Ohrstein–Uhlenbeck processes OIS see overnight index swaps Omega ratio “open” prices option pricing Black–Scholes formula CRR model exotic options finite difference methods implied volatility jump processes Merton model Monte Carlo simulations sensitivities valuation troubles volatility see also prices/pricing options bond duration credit derivative valuation option contract value pricing see also exotic options Organisation for Economic Co-operation and Development (OECD) out of the money (OTM) caps options outright forward operation overnight index swaps (OIS) parametric method, VaR Parkinson volatility participating forward contracts (PFCs) path-dependent options payer swaps percent per annum performance absolute measures attribution Calmar ratio contribution global example IR Jensen’s alpha market MDD non-normal returns Omega ratio relative measures risk measures Sharpe ratio Sortino ratio stocks portfolios swaps TE Treynor ratio Z-score PFCs see participating forward contracts platykurtic distributions POF see Proportion of failures test Poisson processes polynomial curve methods portfolios bond duration bond selection immunization performance attribution contribution Portfolio Theory risk management Portfolio Theory APT model CAPM equities hypotheses Markowitz model performance risk and return valuation troubles “position risk” concept present value (PV) bond duration CRSs IRSs short-term rates spot rates zero-coupon swaps price of risk, CAPM prices/pricing APT model bid/ask bonds CAPM caps CBs CDOs CRSs floors futures high/low IRSs “open”/“close” second-generation swaps spot instruments swaptions see also market prices; option pricing price of time, CAPM price-weighted indexes pricing sensitivities see sensitivities probability risk neutral see also stochastic processes Proportion of failures (POF) test putable bonds put options call-put parity see also options PV see present value quanto swaps randomness random numbers random walks RaV see Risk at Value realized volatility models real option method receiver swaps recovery rates reference currency (ref) regime-switching models regression, NNs relative VaR return measures expected return performance Portfolio Theory in practice risk vs return ratios several stock positions single stock positions time periods returns general Wiener process instantaneous measures “reverse cash and carry” operations rho risk see individual types Risk at Value (RaV) “risk-free” bonds risk-free yield curve risk management risk measures performance attribution contribution Portfolio Theory return measures risk vs return ratios several positions single position VaR risk neutral probability risk premium, CAPM “risky” bonds Rogers–Satchell volatility Roll, R.

This can be explained by economic factors: a company is supposed to re-invest all or part (in case of dividend distribution) of its profits, and thus grow over time, and stock prices must also follow inflation over the long run. Of course, on a shorter horizon of time, prices may decline, even during periods lasting several consecutive years. So that, equity and index options pricing models clearly fit with the random walk hypothesis (although not necessarily strictly Gaussian). Currency prices do not present any global trend over time: a currency is priced relatively to another currency, and economic as well as speculative hazards comfort the random walk hypothesis. But over time, interest rates show the peculiar behavior of successive rising and falling phases. Unfortunately, there is no hope for anticipating both the amplitude and the periodicity of such cycles. We may carefully bound these cycles by, upwards, the “abnormally” very high (more than, say, 15% p.a.) interest rates around the 1980s (that is, before central banks learned to actually control inflation) and by 0 downwards: since the 1990s, Japan has faced interest rates at 0% or very slightly higher, but actually no negative interest rates, including inflation.

The comparison between a (y + risk premium) and the yield of a risk-less bond of the same features is called yield spread analysis. Beyond its impact of a risky bond price, the credit default risk will be further developed in Chapter 13. Clean Price versus Dirty Price Bond prices are quoted by the market as shown above. But on the secondary market, in the case of a trade between two coupon dates, in addition to the market quoted price the buyer must pay to the seller the portion of the coupon pro rata temporis, called accrued interest. The same principle is also applied in accounting, between two coupon dates, according to the Mark-to-Market rules imposed by IFRS standards. The quoted price, ex coupon, is called the clean price, while the (full) price actually paid is the dirty price. Only the clean price is subject to price changes, due to Eq. 3.3, because of the effect of the discount rate y, subject to market rate changes.


pages: 471 words: 97,152

Animal Spirits: How Human Psychology Drives the Economy, and Why It Matters for Global Capitalism by George A. Akerlof, Robert J. Shiller

"Robert Solow", affirmative action, Andrei Shleifer, asset-backed security, bank run, banking crisis, business cycle, buy and hold, collateralized debt obligation, conceptual framework, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, Deng Xiaoping, Donald Trump, Edward Glaeser, en.wikipedia.org, experimental subject, financial innovation, full employment, George Akerlof, George Santayana, housing crisis, Hyman Minsky, income per capita, inflation targeting, invisible hand, Isaac Newton, Jane Jacobs, Jean Tirole, job satisfaction, Joseph Schumpeter, Long Term Capital Management, loss aversion, market bubble, market clearing, mental accounting, Mikhail Gorbachev, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, mortgage debt, Myron Scholes, new economy, New Urbanism, Paul Samuelson, plutocrats, Plutocrats, price stability, profit maximization, purchasing power parity, random walk, Richard Thaler, Robert Shiller, Robert Shiller, Ronald Reagan, South Sea Bubble, The Chicago School, The Death and Life of Great American Cities, The Wealth of Nations by Adam Smith, too big to fail, transaction costs, tulip mania, working-age population, Y2K, Yom Kippur War

., 178n6 Present value theory, 152–53 Price rigidity, 48 Prices: depression of the 1890s and, 59–60; fairness and, 6, 21, 22; Great Depression and, 68; money illusion and, 43–46, 48; variation in, 100. See also financial prices price-to-earnings-to-price feedback, 135 price-to-GDP-to-price feedback, 154 price-to-price feedback, 134–35, 154 Primary Credit Dealer Facility, 187n10 Princeton University, 19 prohibition (of alcohol), 39 Project Link, 16 Pullman Palace Car Company, 63 Purdue University, 128 Quetzalcóatl (López Portillo), 53–54 Quintini, Glenda, 183n14 quits, wages and, 103–4, 106 railroad strike of 1910 (Argentina), 139 Rainwater, Lee, 162, 196n14 Rajan, Raghuram G., 182n21 Randers, Jørgen, 194n29 randomness, 52 random-walk hypothesis, 103, 191n11 ratings of securities, 37, 91, 94, 170 rational expectations, xxiii, 5, 6, 168, 173, 178n4; bimetallism debate and, 60; in classical economics, 2, 3; confidence and, 12–13, 14; corruption and, 39; fairness and, 21, 22; feedback and, 140; financial prices and, 131, 132, 133, 136; money illusion and, 41, 42; real estate market and, 150, 153; saving and, 120, 122 Reagan, Ronald, xxv, 32, 36, 172, 175 real business cycle models, 178n6 real estate market, 4, 6, 135, 136, 149–56, 169–70, 172, 174, 195–96n1–15; baby boom and, 152; confidence and, 11, 13, 149, 156; confidence multiplier in, 153–55; naïve or intuitive beliefs about, 150–53; S&L crisis and, 32, 33.

Shefrin and Thaler assembled some of the evidence of anomalies and produced a behavioral life-cycle model that incorporates some of the best features of the life-cycle model of Ando and Modigliani (1963) and adjusts it for known facts about human behavior (Thaler 1994). For a broader discussion of these issues see Thaler (1994). 10. Keynes (1973 [1936], p. 96). 11. Hall (1978) found some apparently striking evidence in favor of this maximizing model in showing that a time series of aggregate U.S. consumption was approximately a random walk. However, subsequent evidence has generated other interpretations (Blinder et al. 1985; Hall 1988). Carroll and Summers (1991) found evidence against the random-walk hypothesis in that individual consumption tends to track predictable life-cycle changes in income, though Carroll (2001) backtracked a bit on their conclusions. Shea (1995a) found evidence that individual consumption changes can be forecast using data on future incomes implicit in union contracts. 12. Modigliani and Brumberg (1954) and Friedman (1957). 13.

If people tend to buy in reaction to stock price increases or sell in reaction to price decreases, then their reaction to past price changes has the potential to feed back into more price changes in the same direction, a phenomenon known as price-to-price feedback.10 A vicious circle can develop, causing a continuation of the cycle, at least for a while. Eventually an upward price movement, a bubble, must burst, since price is supported only by expectations of further price increases. They cannot go on forever. Price-to-price feedback itself may not be strong enough to create the major asset price bubbles we have seen. But, as we shall see, there are other forms of feedback besides that between prices. In particular there are feedbacks between the asset prices in the bubble and the real economy. This additional feedback increases the length of the cycle and amplifies the price-to-price effects.


pages: 999 words: 194,942

Clojure Programming by Chas Emerick, Brian Carper, Christophe Grand

Amazon Web Services, Benoit Mandelbrot, cloud computing, continuous integration, database schema, domain-specific language, don't repeat yourself, en.wikipedia.org, failed state, finite state, Firefox, game design, general-purpose programming language, Guido van Rossum, Larry Wall, mandelbrot fractal, Paul Graham, platform as a service, premature optimization, random walk, Ruby on Rails, Schrödinger's Cat, semantic web, software as a service, sorting algorithm, Turing complete, type inference, web application

However, we can easily recreate index-step with stepper as well, assuming w and h are globally or locally bound to the width and height of the desired finite grid: (stepper #(filter (fn [[i j]] (and (< -1 i w) (< -1 j h))) (neighbours %)) #{2 3} #{3}) Maze generation Let’s study another example: Wilson’s maze generation algorithm.[112] Wilson’s algorithm is a carving algorithm; it takes a fully walled “maze” and carves an actual maze out of it by removing some walls. Its principle is: Randomly pick a location and mark it as visited. Randomly pick a location that isn’t visited yet—if there’s none, return the maze. Perform a random walk starting from the newly picked location until you stumble on a location that is visited—if you pass through a location more than once during the random walk, always remember the direction you take to leave it. Mark all the locations of the random walk as visited, and remove walls according to the last known “exit direction.” Repeat from 2. Generally, maze algorithms use a matrix to represent the maze, and each item of this matrix is a bitset indicating which walls are still up. The astute reader may twitch at the idea that in such a setup, as the state of a wall is stored twice: once in each location on each side of it.

(iterate (comp rand-nth paths) loc) generates an infinite random walk: it takes a location, applies paths on it to get the vector of adjacent locations and rand-nth to pick one. If paths had returned sets instead of a sequential type (like a vector), then (comp rand-nth seq paths) would have been necessary instead. (take-while unvisited walk) is the part of the random walk until (but not including) a visited location. (take-while unvisited walk) would be (take-while (complement visited) walk) if the code had been written with visited. (next walk) is infinite, but (take-while unvisited walk) is not, so zipmap only looks at the n first items of (next walk) (where n is (count (take-while unvisited walk))). The n first items of (next walk) is thus the random walk without the start location and including the first visited location.

"Maze") (.setContentPane (doto (proxy [javax.swing.JPanel] [] (paintComponent [^java.awt.Graphics g] (let [g (doto ^java.awt.Graphics2D (.create g) (.scale 10 10) (.translate 1.5 1.5) (.setStroke (java.awt.BasicStroke. 0.4)))] (.drawRect g -1 -1 w h) (doseq [[[xa ya] [xb yb]] (map sort maze)] (let [[xc yc] (if (= xa xb) [(dec xa) ya] [xa (dec ya)])] (.drawLine g xa ya xc yc)))))) (.setPreferredSize (java.awt.Dimension. (* 10 (inc w)) (* 10 (inc h)))))) .pack (.setVisible true))) (draw 40 40 (maze (grid 40 40))) The True Wilson’s Algorithm Actually, we fibbed: maze is not exactly an implementation of Wilson’s algorithm. In our code, when the random walk reaches a location already in the graph (maze), we add a whole tree constituted by all the locations visited during the random walk instead of just the branch of this tree going from the starting point to the end location. Obviously, our algorithm is faster since it adds more locations to the maze at once. However, the selling point of Wilson’s algorithm is that each maze has the same probability to be generated. Empirical measures (looking at the maze distribution over samples generated by both Wilson’s and our algorithms) hint that this property still holds in our variant, but we haven’t proved it formally and we haven’t computed its time complexity.


pages: 117 words: 31,221

Fred Schwed's Where Are the Customers' Yachts?: A Modern-Day Interpretation of an Investment Classic by Leo Gough

Albert Einstein, banking crisis, Bernie Madoff, corporate governance, discounted cash flows, diversification, fixed income, index fund, Long Term Capital Management, Northern Rock, passive investing, Ralph Waldo Emerson, random walk, short selling, South Sea Bubble, The Nature of the Firm, the rule of 72, The Wealth of Nations by Adam Smith, transaction costs, young professional

A series of outcomes where the odds have been 50\50 each time constitutes a truly random sequence - there is no hidden pattern to it. In the short-term, according to many statisticians, share prices tend to approximate this level of randomness. There is really no good reason why a price goes up a tick one minute and down a tick the next minute. There is even a theory, called the Random Walk theory, that attempts to explain much of how share prices change in these terms. Over longer periods, though, share price movements look a lot less random. Companies that are doing well in the real world, for instance, tend to enjoy substantial rises in their share price as investors become willing to pay higher prices for the shares. The great question is whether the price of a share is fair, a bargain, or expensive. Nobody really knows what the true price should be because this is subject to market forces, which are almost impossible to measure in most real world situations.

L. 44 mergers and acquisitions (M&As) 88–9 middle age 99 Milken, Michael 73 millionaires, behaviour of 46 Minogue, Kenneth 24 misinformation 72–3, 90, 91 Modern Portfolio Theory 49 modernity and globalisation 78–9 momentum 38–9 mutual funds 86 average returns 105 N Nabisco 89 ‘Names’ 45 new issues 56–7 newsletters 96–7 O Ogilvy, David 88 online brokers 71 ‘open-end’ funds 86 optimism 42–3 ‘options’ 32 OTC (Over The Counter) market 70 overseas markets and diversification 49 P passive investment 111 patterns, identifying 40–1 Patton, George 80 pension schemes 99 Pope, John 104 popular investments 30–1 positive news about companies 42–3 predicting by analysts 97 the market 40–1 returns 80–1 small changes 38–9 ‘present value’ 94 price/book ratio 92 price/earnings (P/E) ratio 55 price/equity to growth (PEG) ratio 93 price/sales ratio 93 probability 26–7 professional stock-pickers 20–1 professionals 96, 98 commissions 72 and investment skills 47, 54–5 risk aversion 58 profit and company size 105 hiding 91 prospectuses 35, 56–7, 87 public overspending 28–9 R Random Walk theory 27 ratio price/book 92 price/earnings (P/E) 55 price/equity to growth (PEG) 93 price/sales 93 reading about stock markets 112–13 ‘regression to the mean’ 105 regulators 84–5 retirement 99 planning for 60–1 returns and diversification 49 estimating 80–1 inflation-adjusted 103 on investments 58–9 long term 64–5 risk adjustment 80–1 and hedge funds 100–1 in large investments 44–5 perception of 108 relative 50–2 in short selling 82–3 in speculation 12–13 and variance 81 Rogers, Jim 36, 77 Rogers, Will 58 Rowe, David 38 Royal Mail 66–7 ‘rule of 72’ 103 rumours, affecting the market 83 Russia, government bonds 51 S Sarbanes-Oxley Act (2002) 85 Schwed, Fred, background 8–9 SEC (Securities and Exchange Commission) 14, 84 Seneca 46 Seven (bank) 41 Shakespeare, William 86 share analysts 37 shares popular 30–1 prices 14–15 releasing 25 short sellers 82–3 short-term fluctuations in share prices 38–9 short-term investing 63 size of company and profit 105 Soros, George 77, 101 South Sea Bubble 56, 109 speculation 12–13 spread betting companies 83 stages of life, planning for 98–9 start-up businesses 106–7 Steinherr, Alfred 20, 33 stock indices 34–5 stock markets share prices 14–15 speculation in 12–13 stock-picking 20–1 stockbrokers 71 T Taiwan 87 takeovers 88–9 tax on trust funds 58–9 technical analysis (TA) 40–1, 112 telecoms companies 108 Thinc Group 84 timing of investments 64–5 tracker funds 21, 62–3 tracking error 63 ‘traction’ 38–9 traders, investment skills 47 see also professionals transaction costs 70–1 trusts 35, 48–9 low returns 58–9 turnarounds 66–7 Twain, Mark 12, 14 U ‘unit trusts’ 86 US, government bonds 80 V Vanderbilt family 16 variance 81 volatility in the Far East 76–7 and risk 51–2, 58 W Wall Street crash 82, 104 Walsh, David 73 wealth passing down through generations 16–17 and relative risk 44–5 and skill in investment 46–7 Wells, H.

On large stock exchanges, like London and New York, millions of shares exchange hands every day, and the share prices are always changing (usually by only a tiny fraction). But in order to sell your shares, there has to be a buyer, and if a government stepped in to force prices up artificially, there would come a point when there would be no buyers: the price would be too high. People would start looking for other ways to invest and billions would drain out of the market and go overseas to other stock exchanges. To function properly, a stock market has to allow prices to fall as well as rise. That’s what investors really want. If Megaboom Plc loses billions and comes near to bankruptcy, you would want to see those losses reflected in its share price, wouldn’t you? If you wouldn’t, then don’t become a stock market investor! But why would anyone sensible buy Megaboom’s shares if they dropped?


pages: 589 words: 69,193

Mastering Pandas by Femi Anthony

Amazon Web Services, Bayesian statistics, correlation coefficient, correlation does not imply causation, Debian, en.wikipedia.org, Internet of things, natural language processing, p-value, random walk, side project, statistical model, Thomas Bayes

The exponential distribution can be described as the continuous limit of the Geometric distribution and is also Markovian (memoryless). A memoryless random variable exhibits the property whereby its future state depends only on relevant information about the current time and not the information from further in the past. An example of modeling a Markovian/memoryless random variable is modeling short-term stock price behavior and the idea that it follows a random walk. This leads to what is called the Efficient Market hypothesis in Finance. For more information, refer to http://en.wikipedia.org/wiki/Random_walk_hypothesis. The PDF of the exponential distribution is given by =. The expectation and variance are given by the following expression: For a reference, refer to the link at http://en.wikipedia.org/wiki/Exponential_distribution. The plot of the distribution and code is given as follows: In [15]: import scipy.stats clrs = colors.cnames x = np.linspace(0,4, 100) expo = scipy.stats.expon lambda_ = [0.5, 1, 2, 5] plt.figure(figsize=(12,4)) for l,c in zip(lambda_,clrs): plt.plot(x, expo.pdf(x, scale=1.

probability density function (PDF) / Continuous probability distributions probability distributionsabout / Probability distributions probability mass function (pmf)about / Discrete probability distributions PYMC Pandas ExampleURL / IPython Notebook PyPI Readline packageURL / Windows Pythonabout / How Python and pandas fit into the data analytics mix features / How Python and pandas fit into the data analytics mix URL / How Python and pandas fit into the data analytics mix, Selecting a version of Python to use, Installing Python from compressed tarball libraries / How Python and pandas fit into the data analytics mix version, selecting / Selecting a version of Python to use installation, on Linux / Linux installation, on Windows / Core Python installation installation, on Mac OS/X / Mac OS X Anaconda package, URL / Installation of Python and pandas from a third-party vendor Python(x,y)URL / Other numeric or analytics-focused Python distributions Python 3.0URL / Selecting a version of Python to use references / Selecting a version of Python to use Python decoratorsreference link / pandas/util Python dictionary, DataFrame objectsDataFrame.to_panel method, using / Using the DataFrame.to_panel method DataFrame.to_panel method, references / Using the DataFrame.to_panel method other operations / Other operations Python extensionsused, for improving performance / Improving performance using Python extensions Python installation, on Linuxabout / Linux from compressed tarball / Installing Python from compressed tarball Python installation, on Mac OS/Xabout / Mac OS X URL / Mac OS X package manager, using / Installation using a package manager Python installation, on Windowsabout / Windows core Python installation / Core Python installation third-party software install / Third-party Python software installation URL / Third-party Python software installation Python Lexical AnalysisURL / Accessing attributes using dot operator Q quartileabout / Quartile reference link / Quartile R Rdata types / R data types column name, specifying in / Specifying column name in R multiple columns, selecting in / Multicolumn selection in R %in% operator / R %in% operator logical subsetting / Logical subsetting in R split-apply-combine, implementing in / Implementation in R melt() function / The R melt() function cut() method / An R example using cut() R, and pandasmatching operators, comparing in / Comparing matching operators in R and pandas R-matrixversus Numpy array / R-matrix and NumPy array compared random forest / Random forest random walk hypothesisreference link / The exponential distribution range / Range R DataFramesabout / R DataFrames versus pandas DataFrames / R's DataFrames versus pandas' DataFrames README file, scikit-learnreference link / Installing on Windows R listsabout / R lists versus pandas series / R lists and pandas series compared role of pandas, in machine learning / Role of pandas in machine learning S sample covariancereference link / The mean sample meanreference link / The mean scikit-learnabout / Role of pandas in machine learning installing / Installation of scikit-learn installing, via Anacondas / Installing via Anaconda installing, on Unix (Linux/Mac OSX) / Installing on Unix (Linux/Mac OS X) installing, on Windows / Installing on Windows reference link / Installing on Windows model. constructing for / Constructing a model using Patsy for scikit-learn scikit-learn ML/classifier interfaceabout / The scikit-learn ML/classifier interface reference link / The scikit-learn ML/classifier interface scipy.stats functionreference link / Quartile Scipy Lecture Notes, Interfacing with Creference link / Improving performance using Python extensions Seriescreating / Series creation creating, with numpy.ndarray / Using numpy.ndarray creating, with Python dictionary / Using Python dictionary creating, with scalar values / Using scalar values operations / Operations on Series Series operationsassignment / Assignment slicing / Slicing arithmetic and statistical operations / Other operations Setuptoolsabout / Third-party Python software installation URL / Third-party Python software installation shape manipulation, NumPy arrayabout / Array shape manipulation multi-dimensional array, flattening / Flattening a multi-dimensional array reshaping / Reshaping resizing / Resizing dimension, adding / Adding a dimension shifting / Shifting/lagging single rowappending, to DataFrame / Appending a single row to a DataFrame sortlevel() method / MultiIndexing sparse.pyreference link / pandas/core split-apply-combineabout / Split-apply-combine implementing, in R / Implementation in R implementing, in pandas / Implementation in pandas SQL-like merging/joining, of DataFrame objects / SQL-like merging/joining of DataFrame objects SQL joinsreference link / SQL-like merging/joining of DataFrame objects stack() functionabout / The stack() function stackingabout / Stacking and unstacking statistical hypothesis testsabout / Statistical hypothesis tests background / Background z-test / The z-test t-test / The t-test structured array, DataFrameURL / Using a structured array submodules, pandas/compatchainmap.py / pandas/compat chainmap_impl.py / pandas/compat pickle_compat.py / pandas/compat openpyxl_compat.py / pandas/compat submodules, pandas/computationapi.py / pandas/computation align.py / pandas/computation common.py / pandas/computation engines.py / pandas/computation eval.py / pandas/computation expressions.py / pandas/computation ops.py / pandas/computation pytables.py / pandas/computation scope.py / pandas/computation submodules, pandas/coreapi.py / pandas/core array.py / pandas/core base.py / pandas/core common.py / pandas/core config.py / pandas/core datetools.py / pandas/core frame.py / pandas/core generic.py / pandas/core categorical.py / pandas/core format.py / pandas/core groupby.py / pandas/core ops.py / pandas/core index.py / pandas/core internals.py / pandas/core matrix.py / pandas/core nanops.py / pandas/core panel.py / pandas/core panel4d.py / pandas/core panelnd.py / pandas/core series.py / pandas/core sparse.py / pandas/core strings.py / pandas/core submodules, pandas/ioapi.py / pandas/io auth.py / pandas/io common.py / pandas/io data.py / pandas/io date_converters.py / pandas/io excel.py / pandas/io ga.py / pandas/io gbq.py / pandas/io html.py / pandas/io json.py / pandas/io packer.py / pandas/io parsers.py / pandas/io pickle.py / pandas/io pytables.py / pandas/io sql.py / pandas/io to_sql(..) / pandas/io stata.py / pandas/io wb.py / pandas/io submodules, pandas/rpybase.py / pandas/rpy common.py / pandas/rpy mass.py / pandas/rpy var.py / pandas/rpy submodules, pandas/sparseapi.py / pandas/sparse array.py / pandas/sparse frame.py / pandas/sparse list.py / pandas/sparse panel.py / pandas/sparse series.py / pandas/sparse submodules, pandas/statsapi.py / pandas/stats common.py / pandas/stats fama_macbeth.py / pandas/stats interface.py / pandas/stats math.py / pandas/stats misc.py / pandas/stats moments.py / pandas/stats ols.py / pandas/stats plm.py / pandas/stats var.py / pandas/stats submodules, pandas/toolsutil.py / pandas/tools tile.py / pandas/tools rplot.py / pandas/tools plotting.py / pandas/tools pivot.py / pandas/tools merge.py / pandas/tools describe.py / pandas/tools submodules, pandas/tseriesapi.py / pandas/tseries converter.py / pandas/tseries frequencies.py / pandas/tseries holiday.py / pandas/tseries index.py / pandas/tseries interval.py / pandas/tseries offsets.py / pandas/tseries period.py / pandas/tseries plotting.py / pandas/tseries resample.py / pandas/tseries timedeltas.py / pandas/tseries tools.py / pandas/tseries util.py / pandas/tseries submodules, pandas/utilterminal.py / pandas/util print_versions.py / pandas/util misc.py / pandas/util decorators.py / pandas/util clipboard.py / pandas/util supervised learningversus unsupervised learning / Supervised versus unsupervised learning about / Supervised learning supervised learning algorithmsabout / Supervised learning algorithms model, constructing for scikit-learn with Patsy / Constructing a model using Patsy for scikit-learn general boilerplate code explanation / General boilerplate code explanation logistic regression / Logistic regression support vector machine (SVM) / Support vector machine decision trees / Decision trees random forest / Random forest supervised learning problemsclassification / Supervised versus unsupervised learning regression / Supervised versus unsupervised learning support vector machine (SVM) / Support vector machineURL / Support vector machine swaplevel function / Swapping and reordering levels SWIG Documentationreference link / Improving performance using Python extensions switchpoint detection, Bayesian analysis example / Bayesian analysis example – Switchpoint detection T t-distributionreference link / The t-test t-testabout / The t-test one sample independent t-test / Types of t-tests independent samples t-tests / Types of t-tests paired samples t-test / Types of t-tests reference link / Types of t-tests example / A t-test example tailed testreference link / Statistical hypothesis tests time-series-related instance methodsabout / Time series-related instance methods shifting/lagging / Shifting/lagging frequency conversion / Frequency conversion data, resampling / Resampling of data aliases, for Time Series frequencies / Aliases for Time Series frequencies Time-Series-related objectsdatetime.datetime / A summary of Time Series-related objects Timestamp / A summary of Time Series-related objects DatetimeIndex / A summary of Time Series-related objects Period / A summary of Time Series-related objects PeriodIndex / A summary of Time Series-related objects DateOffset / A summary of Time Series-related objects timedelta / A summary of Time Series-related objects TimeDelta object / DateOffset and TimeDelta objects time serieshandling / Handling time series TimeSeries.resample functionabout / Resampling of data Time series conceptsabout / Time series concepts and datatypes time series datareading in / Reading in time series data TimeDelta object / DateOffset and TimeDelta objects DateOffset object / DateOffset and TimeDelta objects Time series datatypesabout / Time series concepts and datatypes Period / Period and PeriodIndex PeriodIndex / PeriodIndex Time Series datatypesconversion between / Conversions between Time Series datatypes time series datatypesPeriodIndex / PeriodIndex Time Series frequenciesaliases / Aliases for Time Series frequencies Titanic problemnaïve approach / A naïve approach to Titanic problem transform() method / The transform() method Type I Error / Type I and Type II errors Type II Error / Type I and Type II errors U UEFA Champions LeagueURL / The groupby operation unbiased estimatorreference link / Deviation and variance Unix (Linux/Mac OSX)scikit-learn, installing on / Installing on Unix (Linux/Mac OS X) unstackingabout / Stacking and unstacking unsupervised learningversus supervised learning / Supervised versus unsupervised learning about / Unsupervised learning unsupervised learning algorithmsabout / Unsupervised learning algorithms dimensionality reduction / Dimensionality reduction K-means clustering / K-means clustering upsamplingabout / Resampling of data V 4V’s of big dataabout / 4 V's of big data, Veracity of big data volume / Volume of big data velocity / Velocity of big data variety / Variety of big data veracity / Veracity of big data varianceabout / Deviation and variance variety, big data / Variety of big data vector auto-regression classes, var.pyVAR / pandas/stats PanelVAR / pandas/stats vector autoregressionreference link / pandas/stats velocity, big data / Velocity of big data veracity, big data / Veracity of big data virtualenv toolabout / Virtualenv installing / Virtualenv installation and usage using / Virtualenv installation and usage URL / Virtualenv installation and usage volume, big data / Volume of big data W Wakariabout / Wakari by Continuum Analytics URL / Wakari by Continuum Analytics where() method / Using the where() method WindowsPython, installing / Windows, Core Python installation Anaconda installation / Windows panda installation / Windows IPython installation / Windows scikit-learn, installing on / Installing on Windows WinPythonURL / Other numeric or analytics-focused Python distributions World Bank Economic dataURL / Benefits of using pandas X xs method / Cross sections Z z-testabout / The z-test zettabytesURL / Volume of big data

Multicolumn selection in R In R, we specify the multiple columns to select by stating them in a vector within square brackets: >stocks_table[c('Symbol','Price')] Symbol Price 1 GOOG 518.70 2 AMZN 307.82 3 FB 74.90 4 AAPL 109.70 5 TWTR 37.10 6 NFLX 334.48 7 LINKD 219.90 >stocks_table[,c('Symbol','Price')] Symbol Price 1 GOOG 518.70 2 AMZN 307.82 3 FB 74.90 4 AAPL 109.70 5 TWTR 37.10 6 NFLX 334.48 7 LINKD 219.90 Multicolumn selection in pandas In pandas, we subset elements in the usual way with the column names in square brackets: In [140]: stocks_df[['Symbol','Price']] Out[140]:Symbol Price 0 GOOG 518.70 1 AMZN 307.82 2 FB 74.90 3 AAPL 109.70 4 TWTR 37.10 5 NFLX 334.48 6 LNKD 219.90 In [145]: stocks_df.loc[:,['Symbol','Price']] Out[145]: Symbol Price 0 GOOG 518.70 1 AMZN 307.82 2 FB 74.90 3 AAPL 109.70 4 TWTR 37.10 5 NFLX 334.48 6 LNKD 219.90 Arithmetic operations on columns In R and pandas, we can apply arithmetic operations in data columns in a similar manner.


Think OCaml by Nicholas Monje, Allen Downey

en.wikipedia.org, random walk

For example, reading your code might help if the problem is a typographical error, but not if the problem is a conceptual misunderstanding. If you don’t understand what your program does, you can read it 100 times and never see the error, because the error is in your head. Running experiments can help, especially if you run small, simple tests. But if you run experiments without thinking or reading your code, you might fall into a pattern I call “random walk programming,” which is the process of making random changes until the program does the right thing. Needless to say, random walk programming can take a long time. You have to take time to think. Debugging is like an experimental science. You should have at least one hypothesis about what the problem is. If there are two or more possibilities, try to think of a test that would eliminate one of them. Taking a break helps with the thinking. So does talking. If you explain the problem to someone else (or even yourself), you will sometimes find the answer before you finish asking the question.

Index abecedarian, 51 access, 56 accumulator histogram, 103 Ackerman function, 42 addition with carrying, 46 algorithm, 4, 9, 46, 107 Euclid, 43 RSA, 84 square root, 47 ambiguity, 6 anagram, 61 anagram set, 94 and operator, 32 Anonymous functions, 59 argument, 21, 23, 25, 29 arithmetic operator, 13 assignment, 18 tuple, 87–89, 94 assignment statement, 12 base case, 42 benchmarking, 109, 110 binary search, 61 bingo, 94 birthday paradox, 61 bisection search, 61 bisection, debugging by, 47 body, 23, 29, 35 boolean expression, 31, 35 borrowing, subtraction with, 46 bracket operator, 49 branch, 35 bug, 4, 9 calculator, 19 caml.inria.fr, 10 Car Talk, 85, 95 carrying, addition with, 46 case-sensitivity, variable names, 17 chained conditional, 33 char type, 11 character, 18, 49 comment, 17, 19 comparison string, 52 tuple, 89 compile, 1, 8 composition, 22, 25, 29 compound statement, 35 concatenation, 19, 26, 51 list, 56 condition, 35 conditional chained, 33 nested, 35 conditional execution, 32 conditional statement, 32, 35 cons operator, 55 consistency check, 84 counter, 53, 78 cummings, e. e., 4 Currying, 27 data structure, 94, 109 debugging, 4, 8, 9, 17, 28, 41, 53, 60, 84, 94, 109 by bisection, 47 emotional response, 8 experimental, 5 declaration, 85 default value, 106, 110 definition function, 22 recursive, 95 deterministic, 101, 110 development plan random walk programming, 110 diagram stack, 26 state, 12 dictionary lookup, 81 looping with, 80 reverse lookup, 81 Index Directive, 2 divisibility, 31 documentation, 10 dot notation, 29 Doyle, Arthur Conan, 5 Doyle, Sir Arthur Conan, 103 DSU pattern, 94 duplicate, 61, 85 element, 55, 60 emotional debugging, 8 empty list, 55 empty string, 53 encapsulation, 46 encryption, 84 epsilon, 46 error runtime, 4, 17, 39 semantic, 4, 12, 17 syntax, 4, 17 error message, 4, 8, 12, 17 escape character, 7 Euclid’s algorithm, 43 evaluate, 14 exception, 4, 9, 17 IndexError, 50 RuntimeError, 39 SyntaxError, 22 TypeError, 49 ValueError, 88 executable, 2, 9 experimental debugging, 5, 110 expression, 13, 14, 19 boolean, 31, 35 Fermat’s Last Theorem, 36 fibonacci function, 82 filter pattern, 60 find function, 51 flag, 85 float type, 11 floating-point, 18, 46 flow of execution, 24, 29 For loop, 70 for loop, 57 formal language, 5, 9 frame, 26 frequency, 79 letter, 94 word, 101, 111 function, 22, 28 113 ack, 42 fibonacci, 82 find, 51 log, 21 randint, 61 recursive, 37 sqrt, 22 String.length, 50 zip, 88 function argument, 25 function call, 21, 29 function definition, 22, 24, 29 function frame, 26 function parameter, 25 function, math, 21 function, reasons for, 28 function, trigonometric, 21 function, tuple as return value, 88 Functional Programming, 7 Functions Anonymous, 59 Currying, 27 gather, 94 GCD (greatest common divisor), 43 global variable, 85 greatest common divisor (GCD), 43 Guarded Patterns, 35 hash function, 85 hashtable, 77, 78, 84, 85 hashtbale subtraction, 106 header, 23, 29 Hello, World, 7 high-level language, 1, 8 Higher-Order Functions, 25 histogram, 79, 85 random choice, 102, 107 word frequencies, 102 HOF, 25 Holmes, Sherlock, 5 homophone, 86 if statement, 32 immutability, 53 implementation, 79, 85, 109 in, 15 index, 49, 53, 56, 77 starting at zero, 49 IndexError, 50 infinite recursion, 39, 42 114 int type, 11 integer, 18 long, 83 interactive mode, 2, 9 interlocking words, 61 interpret, 1, 8 invocation, 53 item, 53, 55 hashtable, 84 item update, 58 key, 77, 84 key-value pair, 77, 84 keyboard input, 33 keyword, 13, 19 labelled parameter, 104 language formal, 5 high-level, 1 low-level, 1 natural, 5 programming, 1 safe, 4 let, 15 letter frequency, 94 letter rotation, 53, 85 Linux, 5 list, 55, 60 concatenation, 56 element, 56 empty, 55 nested, 55 of tuples, 89 operation, 56 traversal, 57, 60 literalness, 6 local variable, 26, 29 log function, 21 logarithm, 111 logical operator, 31, 32 long integer, 83 lookup, 85 lookup, dictionary, 81 loop for, 57 Looping, 70 looping with dictionaries, 80 low-level language, 1, 8 map pattern, 60 Index mapping, 108 Markov analysis, 107 mash-up, 108 math function, 21 McCloskey, Robert, 51 membership binary search, 61 bisection search, 61 hashtable, 78 set, 78 memo, 82, 85 metathesis, 94 method, 53 string, 53 module, 7, 29 pprint, 84 random, 61, 102 string, 101 modulus operator, 31, 35 natural language, 5, 9 nested conditional, 33, 35 nested list, 55, 60 Newton’s method, 45 not operator, 32 number, random, 101 object code, 2, 9 operand, 13, 19 operator, 13, 19 and, 32 bracket, 49 cons, 55 logical, 31, 32 modulus, 31, 35 not, 32 or, 32 overloading, 16 relational, 32 string, 16 operator, arithmetic, 13 optional parameter, 104 or operator, 32 order of operations, 16, 18 override, 110 palindrome, 42 parameter, 25, 26, 29 labelled, 104 optional, 104 parentheses empty, 23 Index matching, 4 overriding precedence, 16 parameters in, 25 tuples in, 87 parse, 6, 9 Partial Application, 27 pattern filter, 60 map, 60 reduce, 60 search, 52, 53 swap, 87 Pattern Matching, 34 Pattern-Matching Guarded, 35 PEMDAS, 16 pi, 48 plain text, 101 poetry, 6 portability, 1, 8 pprint module, 84 precedence, 19 precondition, 61 prefix, 108 pretty print, 84 print statement, 7, 9 problem solving, 1, 8 program, 3, 9 Programming Functional, 7 programming language, 1 Programming Paradigms, 7 Functional, 7 Object-Oriented, 7 Project Gutenberg, 101 prompt, 2, 9, 34 prose, 6 pseudorandom, 101, 110 Puzzler, 85, 95 quotation mark, 7, 11 radian, 21 Ramanujan, Srinivasa, 48 randint function, 61 random function, 102 random module, 61, 102 random number, 101 random text, 108 random walk programming, 110 Read functions, 33 115 Recursion Tail-end, 40 recursion, 37, 42 infinite, 39 traversal, 50 recursive definition, 95 reduce pattern, 60 reducible word, 86, 95 redundancy, 6 References, 15, 69 relational operator, 32 return value, 21, 29 tuple, 88 reverse lookup, dictionary, 81 reverse lookup, hashtable, 85 reverse word pair, 61 rotation letters, 85 rotation, letter, 53 RSA algorithm, 84 rules of precedence, 16, 19 running pace, 19 runtime error, 4, 17, 39 RuntimeError, 39 safe language, 4 sanity check, 84 scaffolding, 84 scatter, 94 Scope, 15 scope, 15 Scrabble, 94 script, 2, 9 script mode, 2, 9 search pattern, 52, 53 search, binary, 61 search, bisection, 61 semantic error, 4, 9, 12, 17 semantics, 4, 9 sequence, 49, 53, 55, 87 set anagram, 94 set membership, 78 shape, 94 sine function, 21 slice, 53 source code, 2, 8 sqrt function, 22 square root, 45 stack diagram, 26 state diagram, 12, 18 116 statement, 18 assignment, 12 conditional, 32, 35 for, 57 if, 32 print, 7, 9 Strictly typed, 14 string, 11, 18 comparison, 52 operation, 16 string method, 53 String module, 50 string module, 101 string type, 11 String.length function, 50 structure, 6 subexpressions, 14 subtraction hashtable, 106 with borrowing, 46 suffix, 108 swap pattern, 87 syntax, 4, 9 syntax error, 4, 9, 17 SyntaxError, 22 Tail-end Recursion, 40 tail-end recursion, 42 testing interactive mode, 2 text plain, 101 random, 108 token, 6, 9 Toplevel, 2 toplevel, 9 traceback, 39, 41 traversal, 50, 52, 53, 80, 89, 103 list, 57 triangle, 36 trigonometric function, 21 tuple, 87, 88, 94 assignment, 87 comparison, 89 tuple assignment, 88, 89, 94 type, 11, 18 char, 11 float, 11 hashtable, 77 int, 11 list, 55 Index long, 83 str, 11 tuple, 87 unit, 12, 23 TypeError, 49 typographical error, 110 underscore character, 13 uniqueness, 61 unit type, 12, 18, 23 update histogram, 103 item, 58 use before def, 17 User input, 33 value, 11, 18, 85 default, 106 tuple, 88 ValueError, 88 variable, 12, 18 local, 26 Variables References, 15 While loop, 70 word frequency, 101, 111 word list, 78 word, reducible, 86, 95 words.txt, 78 zero, index starting at, 49 zip function, 88 Zipf’s law, 111

., 4 Currying, 27 data structure, 94, 109 debugging, 4, 8, 9, 17, 28, 41, 53, 60, 84, 94, 109 by bisection, 47 emotional response, 8 experimental, 5 declaration, 85 default value, 106, 110 definition function, 22 recursive, 95 deterministic, 101, 110 development plan random walk programming, 110 diagram stack, 26 state, 12 dictionary lookup, 81 looping with, 80 reverse lookup, 81 Index Directive, 2 divisibility, 31 documentation, 10 dot notation, 29 Doyle, Arthur Conan, 5 Doyle, Sir Arthur Conan, 103 DSU pattern, 94 duplicate, 61, 85 element, 55, 60 emotional debugging, 8 empty list, 55 empty string, 53 encapsulation, 46 encryption, 84 epsilon, 46 error runtime, 4, 17, 39 semantic, 4, 12, 17 syntax, 4, 17 error message, 4, 8, 12, 17 escape character, 7 Euclid’s algorithm, 43 evaluate, 14 exception, 4, 9, 17 IndexError, 50 RuntimeError, 39 SyntaxError, 22 TypeError, 49 ValueError, 88 executable, 2, 9 experimental debugging, 5, 110 expression, 13, 14, 19 boolean, 31, 35 Fermat’s Last Theorem, 36 fibonacci function, 82 filter pattern, 60 find function, 51 flag, 85 float type, 11 floating-point, 18, 46 flow of execution, 24, 29 For loop, 70 for loop, 57 formal language, 5, 9 frame, 26 frequency, 79 letter, 94 word, 101, 111 function, 22, 28 113 ack, 42 fibonacci, 82 find, 51 log, 21 randint, 61 recursive, 37 sqrt, 22 String.length, 50 zip, 88 function argument, 25 function call, 21, 29 function definition, 22, 24, 29 function frame, 26 function parameter, 25 function, math, 21 function, reasons for, 28 function, trigonometric, 21 function, tuple as return value, 88 Functional Programming, 7 Functions Anonymous, 59 Currying, 27 gather, 94 GCD (greatest common divisor), 43 global variable, 85 greatest common divisor (GCD), 43 Guarded Patterns, 35 hash function, 85 hashtable, 77, 78, 84, 85 hashtbale subtraction, 106 header, 23, 29 Hello, World, 7 high-level language, 1, 8 Higher-Order Functions, 25 histogram, 79, 85 random choice, 102, 107 word frequencies, 102 HOF, 25 Holmes, Sherlock, 5 homophone, 86 if statement, 32 immutability, 53 implementation, 79, 85, 109 in, 15 index, 49, 53, 56, 77 starting at zero, 49 IndexError, 50 infinite recursion, 39, 42 114 int type, 11 integer, 18 long, 83 interactive mode, 2, 9 interlocking words, 61 interpret, 1, 8 invocation, 53 item, 53, 55 hashtable, 84 item update, 58 key, 77, 84 key-value pair, 77, 84 keyboard input, 33 keyword, 13, 19 labelled parameter, 104 language formal, 5 high-level, 1 low-level, 1 natural, 5 programming, 1 safe, 4 let, 15 letter frequency, 94 letter rotation, 53, 85 Linux, 5 list, 55, 60 concatenation, 56 element, 56 empty, 55 nested, 55 of tuples, 89 operation, 56 traversal, 57, 60 literalness, 6 local variable, 26, 29 log function, 21 logarithm, 111 logical operator, 31, 32 long integer, 83 lookup, 85 lookup, dictionary, 81 loop for, 57 Looping, 70 looping with dictionaries, 80 low-level language, 1, 8 map pattern, 60 Index mapping, 108 Markov analysis, 107 mash-up, 108 math function, 21 McCloskey, Robert, 51 membership binary search, 61 bisection search, 61 hashtable, 78 set, 78 memo, 82, 85 metathesis, 94 method, 53 string, 53 module, 7, 29 pprint, 84 random, 61, 102 string, 101 modulus operator, 31, 35 natural language, 5, 9 nested conditional, 33, 35 nested list, 55, 60 Newton’s method, 45 not operator, 32 number, random, 101 object code, 2, 9 operand, 13, 19 operator, 13, 19 and, 32 bracket, 49 cons, 55 logical, 31, 32 modulus, 31, 35 not, 32 or, 32 overloading, 16 relational, 32 string, 16 operator, arithmetic, 13 optional parameter, 104 or operator, 32 order of operations, 16, 18 override, 110 palindrome, 42 parameter, 25, 26, 29 labelled, 104 optional, 104 parentheses empty, 23 Index matching, 4 overriding precedence, 16 parameters in, 25 tuples in, 87 parse, 6, 9 Partial Application, 27 pattern filter, 60 map, 60 reduce, 60 search, 52, 53 swap, 87 Pattern Matching, 34 Pattern-Matching Guarded, 35 PEMDAS, 16 pi, 48 plain text, 101 poetry, 6 portability, 1, 8 pprint module, 84 precedence, 19 precondition, 61 prefix, 108 pretty print, 84 print statement, 7, 9 problem solving, 1, 8 program, 3, 9 Programming Functional, 7 programming language, 1 Programming Paradigms, 7 Functional, 7 Object-Oriented, 7 Project Gutenberg, 101 prompt, 2, 9, 34 prose, 6 pseudorandom, 101, 110 Puzzler, 85, 95 quotation mark, 7, 11 radian, 21 Ramanujan, Srinivasa, 48 randint function, 61 random function, 102 random module, 61, 102 random number, 101 random text, 108 random walk programming, 110 Read functions, 33 115 Recursion Tail-end, 40 recursion, 37, 42 infinite, 39 traversal, 50 recursive definition, 95 reduce pattern, 60 reducible word, 86, 95 redundancy, 6 References, 15, 69 relational operator, 32 return value, 21, 29 tuple, 88 reverse lookup, dictionary, 81 reverse lookup, hashtable, 85 reverse word pair, 61 rotation letters, 85 rotation, letter, 53 RSA algorithm, 84 rules of precedence, 16, 19 running pace, 19 runtime error, 4, 17, 39 RuntimeError, 39 safe language, 4 sanity check, 84 scaffolding, 84 scatter, 94 Scope, 15 scope, 15 Scrabble, 94 script, 2, 9 script mode, 2, 9 search pattern, 52, 53 search, binary, 61 search, bisection, 61 semantic error, 4, 9, 12, 17 semantics, 4, 9 sequence, 49, 53, 55, 87 set anagram, 94 set membership, 78 shape, 94 sine function, 21 slice, 53 source code, 2, 8 sqrt function, 22 square root, 45 stack diagram, 26 state diagram, 12, 18 116 statement, 18 assignment, 12 conditional, 32, 35 for, 57 if, 32 print, 7, 9 Strictly typed, 14 string, 11, 18 comparison, 52 operation, 16 string method, 53 String module, 50 string module, 101 string type, 11 String.length function, 50 structure, 6 subexpressions, 14 subtraction hashtable, 106 with borrowing, 46 suffix, 108 swap pattern, 87 syntax, 4, 9 syntax error, 4, 9, 17 SyntaxError, 22 Tail-end Recursion, 40 tail-end recursion, 42 testing interactive mode, 2 text plain, 101 random, 108 token, 6, 9 Toplevel, 2 toplevel, 9 traceback, 39, 41 traversal, 50, 52, 53, 80, 89, 103 list, 57 triangle, 36 trigonometric function, 21 tuple, 87, 88, 94 assignment, 87 comparison, 89 tuple assignment, 88, 89, 94 type, 11, 18 char, 11 float, 11 hashtable, 77 int, 11 list, 55 Index long, 83 str, 11 tuple, 87 unit, 12, 23 TypeError, 49 typographical error, 110 underscore character, 13 uniqueness, 61 unit type, 12, 18, 23 update histogram, 103 item, 58 use before def, 17 User input, 33 value, 11, 18, 85 default, 106 tuple, 88 ValueError, 88 variable, 12, 18 local, 26 Variables References, 15 While loop, 70 word frequency, 101, 111 word list, 78 word, reducible, 86, 95 words.txt, 78 zero, index starting at, 49 zip function, 88 Zipf’s law, 111


pages: 402 words: 110,972

Nerds on Wall Street: Math, Machines and Wired Markets by David J. Leinweber

AI winter, algorithmic trading, asset allocation, banking crisis, barriers to entry, Big bang: deregulation of the City of London, business cycle, butter production in bangladesh, butterfly effect, buttonwood tree, buy and hold, buy low sell high, capital asset pricing model, citizen journalism, collateralized debt obligation, corporate governance, Craig Reynolds: boids flock, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, Danny Hillis, demand response, disintermediation, distributed generation, diversification, diversified portfolio, Emanuel Derman, en.wikipedia.org, experimental economics, financial innovation, fixed income, Gordon Gekko, implied volatility, index arbitrage, index fund, information retrieval, intangible asset, Internet Archive, John Nash: game theory, Kenneth Arrow, load shedding, Long Term Capital Management, Machine translation of "The spirit is willing, but the flesh is weak." to Russian and back, market fragmentation, market microstructure, Mars Rover, Metcalfe’s law, moral hazard, mutually assured destruction, Myron Scholes, natural language processing, negative equity, Network effects, optical character recognition, paper trading, passive investing, pez dispenser, phenotype, prediction markets, quantitative hedge fund, quantitative trading / quantitative finance, QWERTY keyboard, RAND corporation, random walk, Ray Kurzweil, Renaissance Technologies, risk tolerance, risk-adjusted returns, risk/return, Robert Metcalfe, Ronald Reagan, Rubik’s Cube, semantic web, Sharpe ratio, short selling, Silicon Valley, Small Order Execution System, smart grid, smart meter, social web, South Sea Bubble, statistical arbitrage, statistical model, Steve Jobs, Steven Levy, Tacoma Narrows Bridge, the scientific method, The Wisdom of Crowds, time value of money, too big to fail, transaction costs, Turing machine, Upton Sinclair, value at risk, Vernor Vinge, yield curve, Yogi Berra, your tax dollars at work

The retranslation of the Russian back to English this time was “The spirit is of willing of but of the flesh is of weak.” 31. The CIA In-Q-Tel venture capitalists are found here: www.inqtel.org/. Part Two Alpha as Life 90 Nerds on Wall Str eet I ndex funds are passive investments; their goal is to deliver a return that matches a benchmark index. The Old Testament of indexing is Burton Malkiel’s classic A Random Walk Down Wall Street, first published in 1973 by W.W. Norton and now in its ninth edition. For typical individual investors, without special access to information, it offers what is likely the best financial advice they will ever get: It is hard to consistently beat the market, especially after fees. A passive strategy will do better in the long run. Of course, no one thinks of oneself as a typical individual investor.

.” — HACKER PROVERB T he beginning of index investing in the 1970s was the result of a convergence of events, one of those ripe apple moments. Institutional investors began to use firms like A.G. Becker to actually compare the total performance of their hired managers with index benchmarks, and found that many of them fell short, especially after the substantial fees the investors were paying. Yale professor Burton Malkiel popularized the academic efficient market arguments in A Random Walk Down Wall Street, writing in 1973, “[We need] a new investment instrument: a no-load, minimummanagement-fee mutual fund that simply buys the hundreds of stocks making up the market averages and does no trading [of securities]. . . . Fund spokesmen are quick to point out, ‘you can’t buy the averages.’ It’s about time the public could.” Computers had gotten to the point where one could be put in an office setting without having to tear out walls and bring in industrialstrength air-conditioning, raised floors for the cables, and special power systems.

Dave Goldberg’s opening address to the conference included an insightful assessment of the state of evolutionary computing, in theory and in practice. Somehow, he managed to work in a story involving his Perils and Pr omise of Evolutionary Computation on Wall Str eet 187 Lithuanian grandmother’s recipe for chicken soup, which began, “First, steal a chicken.” There was no Lithuanian chicken soup at that GECCO, but there were some amazing demonstrations of learning programs. Robot control strategies started out as random walks, and after a few hundred simulated generations, they were moving like R2D2 on a good day. There were novel circuit, network, and even protein designs produced by artificial genetic methods.2 The financial guys, many of whom I recognized from more Wall Street–oriented events, and I were trolling for ideas, people to hire, and software to take home. I found all three, and so, several years later, when the GECCO crowd came to New York to focus on financial applications (and maybe get some of these guys hired), I was invited to give one of the keynote talks on what we had been up to.


pages: 321

Finding Alphas: A Quantitative Approach to Building Trading Strategies by Igor Tulchinsky

algorithmic trading, asset allocation, automated trading system, backtesting, barriers to entry, business cycle, buy and hold, capital asset pricing model, constrained optimization, corporate governance, correlation coefficient, credit crunch, Credit Default Swap, discounted cash flows, discrete time, diversification, diversified portfolio, Eugene Fama: efficient market hypothesis, financial intermediation, Flash crash, implied volatility, index arbitrage, index fund, intangible asset, iterative process, Long Term Capital Management, loss aversion, market design, market microstructure, merger arbitrage, natural language processing, passive investing, pattern recognition, performance metric, popular capitalism, prediction markets, price discovery process, profit motive, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk tolerance, risk-adjusted returns, risk/return, selection bias, sentiment analysis, shareholder value, Sharpe ratio, short selling, Silicon Valley, speech recognition, statistical arbitrage, statistical model, stochastic process, survivorship bias, systematic trading, text mining, transaction costs, Vanguard fund, yield curve

Shaw & Co. 8 design 25–30 automated searches 111–120 backtesting 33–41 case study 31–41 core concepts 3–6 data inputs 4, 25–26, 43–47 evaluation 28–29 expressions 4 flow chart 41 future performance 29–30 horizons 4–50 intraday alphas 219–221 machine learning 121–126 noise reduction 26 optimization 29–30 prediction frequency 27 quality 5 risk-on/risk off alphas 246–247 robustness 89–93 smoothing 54–55, 59–60 triple-axis plan 83–88 universe 26 value 27–30 digital filters 127–128 digitization 7–9 dimensionality 129–132 disclosures 192 distressed assets 202–203 diversification automated searches 118–119 exchange-traded funds 233 portfolios 83–88, 108 DL see deep learning dot (inner) product 63–64 Dow, Charles 7 DPIN see dynamic measure of the probability of informed trading drawdowns 106–107 dual timestamping 78 dynamic measure of the probability of informed trading (DPIN) 214–215 dynamic parameterization 132 early-exercise premium 174 earnings calls 181, 187–188 earnings estimates 184–185 earnings surprises 185–186 efficiency, automated searches 111–113 Index295 efficient markets hypothesis (EMH) 11, 135 ego 19 elegance of models 75 EMH see efficient markets hypothesis emotions 19 ensemble methods 124–125 ensemble performance 117–118 estimation of risk 102–106 historical 103–106 position-based 102–103 shrinkage 131 ETFs see exchange-traded funds Euclidean space 64–66 evaluation 13–14, 28–29 backtesting 13–14, 33–41, 69–76 bias 77–82 bootstrapping 107 correlation 28–29 cutting losses 20–21 data selection 74–75 drawdowns 107 information ratio 28 margin 28 overfitting 72–75 risk 101–110 robustness 89–93 turnover 49–60 see also validation event-driven strategies 195–205 business cycle 196 capital structure arbitrage 204–205 distressed assets 202–203 index-rebalancing arbitrage 203–204 mergers 196–199 spin-offs, split-offs & carve-outs 200–202 exchange-traded funds (ETFs) 223–240 average daily trading volume 239 challenges 239–240 merits 232–233 momentum alphas 235–237 opportunities 235–238 research 231–240 risks 233–235 seasonality 237–238 see also index alphas exit costs 19, 21 expectedness of news 164 exponential moving averages 54 expressions, simple 4 extreme alpha values 104 extrinsic risk 101, 106, 108–109 factor risk heterogeneity 234 factors financial statements 147 to alphas 148 failure modes 84 fair disclosures 192 fair value of futures 223 Fama–French three-factor model 96 familiarity bias 81 feature extraction 130–131 filters 127–128 finance blogs 181–182 finance portals 180–181, 192 financial statement analysis 141–154 balance sheets 143 basics 142 cash flow statements 144– 145, 150–152 corporate governance 146 factors 147–148 fundamental analysis 149–154 growth 145–146 income statements 144 negative factors 146–147 special considerations 147 finite impulse response (FIR) filters 127–128 296Index FIR filters see finite impulse response filters Fisher Transform 91 five-day reversion alpha 55–59 Float Boost 125 forecasting behavioral economics 11–12 computer adoption 7–9 frequencies 27 horizons 49–50 statistical arbitrage 10–11 UnRule 17–21 see also predictions formation of the industry 8–9 formulation bias 80 forward-looking bias 72 forwards 241–249 checklist 243–244 Commitments of Traders report 244–245 instrument groupings 242–243 seasonality 245–246 underlying assets 241–242 frequencies 27 full text analysis 164 fundamental analysis 149–154 future performance 29–30 futures 241–249 checklist 243–244 Commitments of Traders report 244–245 fair value 223 instrument groupings 242–243 seasonality 245–246 underlying assets 241–242 fuzzy logic 126 General Electric 200 generalized correlation 64–66 groupings, futures and forwards 242–243 group momentum 157–158 growth analysis 145–146 habits, successful 265–271 hard neutralization 108 headlines 164 hedge fund betas see risk factors hedge funds, initial 8–9 hedging 108–109 herding 81–82, 190–191 high-pass filters 128 historical risk measures 103–106 horizons 49–50 horizontal mergers 197 Huber loss function 129 humps 54 hypotheses 4 ideas 85–86 identity matrices 65 IIR filters see infinite impulse response filters illiquidity premium 208–211 implementation core concepts 12–13 triple-axis plan 86–88 inaccuracy of models 10–11 income statements 144 index alphas 223–240 index changes 225–228 new entrants 227–228 principles 223–225 value distortion 228–230 see also exchange-traded funds index-rebalancing arbitrage 203–204 industry formation 8–9 industry-specific factors 188–190 infinite impulse response (IIR) filters 127–128 information ratio (IR) 28, 35–36, 74–75 initial hedge funds 8–9 inner product see dot product inputs, for design 25–26 integer effect 138 intermediate variables 115 Index297 intraday data 207–216 expected returns 211–215 illiquidity premium 208–211 market microstructures 208 probability of informed trading 213–215 intraday trading 217–222 alpha design 219–221 liquidity 218–219 vs. daily trading 218–219 intrinsic risk 102–103, 105–106, 109 invariance 89 inverse exchange-traded funds 234 IR see information ratio iterative searches 115 Jensen’s alpha 3 L1 norm 128–129 L2 norm 128–129 latency 46–47, 128, 155–156 lead-lag effects 158 length of testing 75 Level 1/2 tick data 46 leverage 14–15 leveraged exchange-traded funds 234 limiting methods 92–93 liquidity effect 96 intraday data 208–211 intraday trading 218–219 and spreads 51 literature, as a data source 44 look-ahead bias 78–79 lookback days, WebSim 257–258 looking back see backtesting Lo’s hypothesis 97 losses cutting 17–21, 109 drawdowns 106–107 loss functions 128–129 low-pass filters 128 M&A see mergers and acquisitions MAC clause see material adverse change clause MACD see moving average convergence-divergence machine learning 121–126 deep learning 125–126 ensemble methods 124–125 fuzzy logic 126 look-ahead bias 79 neural networks 124 statistical models 123 supervised/unsupervised 122 support vector machines (SVM) 122, 123–124 macroeconomic correlations 153 manual searches, pre-automation 119 margin 28 market commentary sites 181–182 market effects index changes 225–228 see also price changes market microstructure 207–216 expected returns 211–215 illiquidity premium 208–211 probability of informed trading 213–215 types of 208 material adverse change (MAC) clause 198–199 max drawdown 35 max stock weight, WebSim 257 mean-reversion rule 70 mean-squared error minimization 11 media 159–167 academic research 160 categorization 163 expectedness 164 finance information 181–182, 192 momentum 165 novelty 161–162 298Index sentiment 160–161 social 165–166 mergers and acquisitions (M&A) 196–199 models backtesting 69–76 elegance 75 inaccuracy of 10–11 see also algorithms; design; evaluation; machine learning; optimization momentum alphas 155–158, 165, 235–237 momentum effect 96 momentum-reversion 136–137 morning sunshine 46 moving average convergencedivergence (MACD) 136 multiple hypothesistesting 13, 20–21 narrow framing 81 natural gas reserves 246 negative factors, financial statements 146–147 neocognitron models 126 neural networks (NNs) 124 neutralization 108 WebSim 257 newly indexed companies 227–228 news 159–167 academic research 160 categories 163 expectedness 164 finance information 181–182, 192 momentum 165 novelty 161–162 relevance 162 sentiment 160–161 volatility 164–165 NNs see neural networks noise automated searches 113 differentiation 72–75 reduction 26 nonlinear transformations 64–66 normal distribution, approximation to 91 novelty of news 161–162 open interest 177–178 opportunities 14–15 optimization 29–30 automated searches 112, 115–116 loss functions 128–129 of parameter 131–132 options 169–178 concepts 169 open interest 177–178 popularity 170 trading volume 174–177 volatility skew 171–173 volatility spread 174 option to stock volume ratio (O/S) 174–177 order-driven markets 208 ordering methods 90–92 O/S see option to stock volume ratio outliers 13, 54, 92–93 out-of-sample testing 13, 74 overfitting 72–75 data mining 79–80 reduction 74–75, 269–270 overnight-0 alphas 219–221 overnight-1 alphas 219 parameter minimization 75 parameter optimization 131–132 PCA see principal component analysis Pearson correlation coefficients 62–64, 90 peer pressure 156 percent profitable days 35 performance parameters 85–86 Index299 PH see probability of heuristicdriven trading PIN see probability of informed trading PnL see profit and loss pools see portfolios Popper, Karl 17 popularity of options 170 portfolios correlation 61–62, 66 diversification 83–88, 108 position-based risk measures 102–103 positive bias 190 predictions 4 frequency 27 horizons 49–50 see also forecasting price changes analyst reports 190 behavioral economics 11–12 efficient markets hypothesis 11 expressions 4 index changes 225–228 news effects 159–167 relative 12–13, 26 price targets 184 price-volume strategies 135–139 pride 19 principal component analysis (PCA) 130–131 probability of heuristic-driven trading (PH) 214 probability of informed trading (PIN) 213–215 profit and loss (PnL) correlation 61–62 drawdowns 106–107 see also losses profit per dollar traded 35 programming languages 12 psychological factors see behavioral economics put-call parity relation 174 Python 12 quality 5 quantiles approximation 91 quintile distributions 104–105 quote-driven markets 208 random forest algorithm 124–125 random walks 11 ranking 90 RBM see restricted Boltzmann machine real estate investment trusts (REITs) 227 recommendations by analysts 182–183 recurrent neural networks (RNNs) 125 reduction of dimensionality 130–131 of noise 26 of overfitting 74–75, 269–270 of risk 108–109 Reg FD see Regulation Fair Disclosure region, WebSim 256 regions 85–86 regression models 10–11 regression problems 121 regularization 129 Regulation Fair Disclosure (Reg FD) 192 REITs see real estate investment trusts relationship models 26 relative prices 12–13, 26 relevance, of news 162 Renaissance Technologies 8 research 7–15 analyst reports 179–193 automated searches 111–120 backtesting 13–14 300Index behavioral economics 11–12 computer adoption 7–9 evaluation 13–14 exchange-traded funds 231–240 implementation 12–13 intraday data 207–216 machine learning 121–126 opportunities 14–15 perspectives 7–15 statistical arbitrage 10–11 triple-axis plan 83–88 restricted Boltzmann machine (RBM) 125 Reuleaux triangle 70 reversion alphas, five-day 55–59 risk 101–110 arbitrage 196–199 control 108–109 drawdowns 106–107 estimation 102–106 extrinsic 101, 106, 108–109 intrinsic 102–103, 105–106, 109 risk factors 26, 95–100 risk-on/risk off alphas 246–247 risk-reward matrix 267–268 RNNs see recurrent neural networks robustness 89–93, 103–106 rules 17–18 evaluation 20–21 see also algorithms; UnRule Russell 2000 IWM fund 225–226 SAD see seasonal affective disorder scale of automated searches 111–113 search engines, analyst reports 180–181 search spaces, automated searches 114–116 seasonality exchange-traded funds 237–238 futures and forwards 245–246 momentum strategies 157 and sunshine 46 selection bias 77–79, 117–118 sell-side analysts 179–180 see also analyst reports sensitivity tests 119 sentiment analysis 160–161, 188 shareholder’s equity 151 Sharpe ratios 71, 73, 74–75, 221, 260 annualized 97 Shaw, David 8 shrinkage estimators 131 signals analysts report 190, 191–192 cutting losses 20–21 data sources 25–26 definition 73 earnings calls 187–188 expressions 4 noise reduction 26, 72–75 options trading volume 174–177 smoothing 54–55, 59–60 volatility skew 171–173 volatility spread 174 sign correlation 65 significance tests 119 Simons, James 8 simple moving averages 55 simulation backtesting 71–72 WebSim settings 256–258 see also backtesting size factor 96 smoothing 54–55, 59–60 social media 165–166 sources of data 25–26, 43–44, 74–75 automated searches 113–114 see also data sparse principal component analysis (sPCA) 131 Spearman’s rank correlation 90 Index301 special considerations, financial statements 147 spin-offs 200–202 split-offs 200–202 spreads and liquidity 51 and volatility 51–52 stat arb see statistical arbitrage statistical arbitrage (stat arb) 10–11, 69–70 statistical models, machine learning 123 step-by-step construction 5, 41 storage costs 247–248 storytelling 80 subjectivity 17 sunshine 46 supervised machine learning 122 support vector machines (SVM) 122, 123–124 systemic bias 77–80 TAP see triple-axis plan tax efficiency, exchange-traded funds 233 teams 270–271 temporal-based correlation 63–64, 65 theory-fitting 80 thought processes of analysts 186–187 tick data 46 timestamping and bias 78–79 tracking errors 233–234 trades cost of 50–52 crossing effect 52–53 latency 46–47 trend following 18 trimming 92 triple-axis plan (TAP) 83–88 concepts 83–86 implementation 86–88 tuning of turnover 59–60 see also smoothing turnover 49–60 backtesting 35 control 53–55, 59–60 costs 50–52 crossing 52–53 examples 55–59 horizons 49–50 smoothing 54–55, 59–60 WebSim 260 uncertainty 17–18 underlying principles 72–73 changes in 109 understanding data 46 unexpected news 164 universes 26, 85–86, 239–240, 256 UnRule 17–18, 20–21 unsupervised machine learning 122 validation, data 45–46 valuation methodologies 189 value of alphas 27–30 value distortion, indices 228–230 value factors 96 value investing 96, 141 variance and bias 129–130 vendors as a data source 44 vertical mergers 197 volatility and news 164–165 and spreads 51–52 volatility skew 171–173 volatility spread 174 volume of options trading 174–177 price-volume strategies 135–139 volume-synchronized probability of informed trading (VPIN) 215 302Index VPIN see volume-synchronized probability of informed trading weather effects 46 WebSim 253–261 analysis 258–260 backtesting 33–41 data types 255 example 260–261 settings 256–258 weekly goals 266–267 weighted moving averages 55 Winsorization 92–93 Yahoo finance 180 Z-scoring 92

The theory also implies that looking for exploitable patterns in prices, and in other forms of publicly available data, will not lead to strategies in which investors can have confidence, from a statistical perspective. An implication of the EMH is that prices will evolve in a process indistinguishable from a random walk. However, another branch of financial economics has sought to disprove the EMH. Behavioral economics studies market imperfections resulting from investor psychological traits or cognitive biases. Imperfections in the financial markets may be due to overconfidence, overreaction, or other defects in how humans process information. Empirical studies may have had mixed results in aiming to disprove the EMH, but if no investors made any 12 Finding Alphas effort to acquire and analyze information, then prices would not reflect all available information and the market would not be efficient. But that in turn would attract profit-motivated investors to tackle the problem of analyzing the information and trading based on it.

Therefore, if they lose against the informed traders (the first terms in equations 1 and 2 below) and win against the uninformed traders (the second terms in equations 1 and 2 below), their profits are a vH 0.5 1 a v 0, (1) vL 0.5 1 v b 0. (2) b The difference between the ask and bid prices, or the spread (S), is then given by S vH 1 0.5 1 vL 1 0.5 1 . (3) Moreover, by further simplifying the model with the assumption that 0.5 , the spread prices follow a random walk at the intraday level becomes a linear function of the probability of informed trading (π): S vH vL . (4) Intraday Data in Alpha Research213 The importance of Glosten and Milgrom’s results lies in the illiquidity premium principle discussed above, stating that higher spreads increase the expected returns. One of the most detailed analyses of this indirect link between the probability of informed trading in a given stock and the stock’s return was conducted by Easley et al. (2002).


Stock Market Wizards: Interviews With America's Top Stock Traders by Jack D. Schwager

Asian financial crisis, banking crisis, barriers to entry, beat the dealer, Black-Scholes formula, commodity trading advisor, computer vision, East Village, Edward Thorp, financial independence, fixed income, implied volatility, index fund, Jeff Bezos, John Meriwether, John von Neumann, locking in a profit, Long Term Capital Management, margin call, money market fund, Myron Scholes, paper trading, passive investing, pattern recognition, random walk, risk tolerance, risk-adjusted returns, short selling, Silicon Valley, statistical arbitrage, the scientific method, transaction costs, Y2K

JOHN BENDER tion in the formula is that the probabilities of prices being at different levels at the time of the option expiration can be described by a normal curve*—the highest probabilities being for prices that are close to the current level and the probabilities for any price decreasing the further above or below the market it is.] A normal distribution would be appropriate if stock price movements were analogous to what is commonly called "the drunkard's walk." If you have a drunkard in a narrow corridor, and all he can do is lurch forward or backward, in order for his movements to be considered a random walk, the following criteria would have to be met: 1. He has to be equally likely to lurch forward as backward. 2. He has to lurch forward by exactly the same distance he lurches backward. 3. He has to lurch once every constant time interval. Those are pretty strict requirements. Not many variables meet these conditions. Stock prices, I would argue, don't even come close [substituting daily price changes for the drunkard's steps], I don't mean to suggest that Black and Scholes made stupid assumptions; they made the only legitimate assumptions possible, not being traders themselves.

For example, if people expect a stock to find support at 65, lo and behold, they're willing to buy it at 66. That is not a random walk statement. *See note in final section of this chapter. Q U E S T I O N I N G THE OBVIOUS I'll give you another example. Assume people get excited about tech stocks for whatever reason and start buying them. Which funds are going to have the best performance next quarter when mom-andpop public decide where to invest their money?—the tech funds. Which funds are going to have the best inflows during the next quarter?—the tech funds. What stocks are they going to buy?—not airlines, they're tech funds. So the tech funds will go up even more. Therefore they're going to have better performance and get the next allocation, and so on. You have all the ingredients for a trend. Again, this is not price behavior that is consistent with a random walk assumption. You've seen this pattern increasingly in the recent run-up in the U.S. stock market.

Stock prices, I would argue, don't even come close [substituting daily price changes for the drunkard's steps], I don't mean to suggest that Black and Scholes made stupid assumptions; they made the only legitimate assumptions possible, not being traders themselves. In fact, they won the Nobel Prize for it. Although, to be honest, that always seemed a bit strange to me because all they used was high school mathematics. All my trading operates on the premise that the most important part is the part that Black-Scholes left out—the assumption of the probability distribution. Why do you say with such assurance that stock prices don't even come close to a random walk? As one example, whether you believe in it or not, there is such a thing as technical analysis, which tries to define support and resistance levels and trends. Regardless of whether technical analysis has any validity, enough people believe in it to impact the market.


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Other People's Money: Masters of the Universe or Servants of the People? by John Kay

Affordable Care Act / Obamacare, asset-backed security, bank run, banking crisis, Basel III, Bernie Madoff, Big bang: deregulation of the City of London, bitcoin, Black Swan, Bonfire of the Vanities, bonus culture, Bretton Woods, buy and hold, call centre, capital asset pricing model, Capital in the Twenty-First Century by Thomas Piketty, cognitive dissonance, corporate governance, Credit Default Swap, cross-subsidies, dematerialisation, disruptive innovation, diversification, diversified portfolio, Edward Lloyd's coffeehouse, Elon Musk, Eugene Fama: efficient market hypothesis, eurozone crisis, financial innovation, financial intermediation, financial thriller, fixed income, Flash crash, forward guidance, Fractional reserve banking, full employment, George Akerlof, German hyperinflation, Goldman Sachs: Vampire Squid, Growth in a Time of Debt, income inequality, index fund, inflation targeting, information asymmetry, intangible asset, interest rate derivative, interest rate swap, invention of the wheel, Irish property bubble, Isaac Newton, John Meriwether, light touch regulation, London Whale, Long Term Capital Management, loose coupling, low cost airline, low cost carrier, M-Pesa, market design, millennium bug, mittelstand, money market fund, moral hazard, mortgage debt, Myron Scholes, NetJets, new economy, Nick Leeson, Northern Rock, obamacare, Occupy movement, offshore financial centre, oil shock, passive investing, Paul Samuelson, peer-to-peer lending, performance metric, Peter Thiel, Piper Alpha, Ponzi scheme, price mechanism, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, railway mania, Ralph Waldo Emerson, random walk, regulatory arbitrage, Renaissance Technologies, rent control, risk tolerance, road to serfdom, Robert Shiller, Robert Shiller, Ronald Reagan, Schrödinger's Cat, shareholder value, Silicon Valley, Simon Kuznets, South Sea Bubble, sovereign wealth fund, Spread Networks laid a new fibre optics cable between New York and Chicago, Steve Jobs, Steve Wozniak, The Great Moderation, The Market for Lemons, the market place, The Myth of the Rational Market, the payments system, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Tobin tax, too big to fail, transaction costs, tulip mania, Upton Sinclair, Vanguard fund, Washington Consensus, We are the 99%, Yom Kippur War

Interest rates are expected to rise, Procter and Gamble owns many powerful brands, the Chinese economy is growing rapidly: these factors are fully reflected in the current level of long-term interest rates, the Procter & Gamble stock price and the exchange rate between the dollar and the renminbi. Since everything that is already known is ‘in the price’, only things that are not already known can influence the price. In an efficient market prices will therefore follow what is picturesquely described as a ‘random walk’ – the next move is as likely to be up as down. And since everything that is known is in the price, that price will represent the best available estimate of the underlying value of a security. A small further step of analogous reasoning leads to the ‘no arbitrage’ condition: each security is appropriately priced in relation to all other securities, so that it is never possible to make money by selling one and buying another.

., 2011, Money and Power: How Goldman Sachs Came to Rule the World, New York, Random House, p. 515. 17. Ceresney, A., 2013, ‘Statement on the Tourre Verdict’, US Securities and Exchange Commission Public Statement, 1 August. 18. Loewenstein, G., 1987, ‘Anticipation and the Value of Delayed Consumption’, Economic Journal, 97 (387), September, pp. 666–84. 19. There are many studies of this. See, for example, Malkiel, B. G., 2012, A Random Walk down Wall Street, 10th edn, New York and London, W.W. Norton. pp. 177–83. Porter, G.E., and Trifts, J.W., 2014, ‘The Career Paths of Mutual Fund Managers: The Role of Merit’, Financial Analysts Journal, 70 (4), July/August, pp. 55–71. Philips, C.B., Kinniry Jr, F.M., Schlanger, T., and Hirt, J.M., 2014, ‘The Case for Index-Fund Investing’, Vanguard Research, April, https://advisors.vanguard.com/VGApp/iip/site/advisor/researchcommentary/article/IWE_InvComCase4Index. 20.

, Proceedings, Federal Reserve Bank of Chicago, pp. 639–45. Loewenstein, G., 1987, ‘Anticipation and the Value of Delayed Consumption’, Economic Journal, 97 (387), September. Lucas Jr, R.E., 2003, ‘Macroeconomic Priorities’, The American Economic Review, 93 (1), March, pp. 1–14. Macmillan, H., 1957, ‘Leader’s Speech’, remarks at Conservative Party rally, Bedford, 20 July. Malkiel, B.G., 2012, A Random Walk down Wall Street, 10th edn, New York and London, W.W. Norton. Manne, H.G., 1965, ‘Mergers and the Market for Corporate Control’, The Journal of Political Economy, 73 (2), April, pp. 110–20. Markopolos, H., 2010, No One Would Listen: A True Financial Thriller, Hoboken, NJ, Wiley. Martin, F., 2013, Money: The Unauthorised Biography, London, Bodley Head. McArdle, M., 2009, ‘Why Goldman Always Wins’, The Atlantic, 1 October.


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Extreme Money: Masters of the Universe and the Cult of Risk by Satyajit Das

affirmative action, Albert Einstein, algorithmic trading, Andy Kessler, Asian financial crisis, asset allocation, asset-backed security, bank run, banking crisis, banks create money, Basel III, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Big bang: deregulation of the City of London, Black Swan, Bonfire of the Vanities, bonus culture, Bretton Woods, BRICs, British Empire, business cycle, capital asset pricing model, Carmen Reinhart, carried interest, Celtic Tiger, clean water, cognitive dissonance, collapse of Lehman Brothers, collateralized debt obligation, corporate governance, corporate raider, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, debt deflation, Deng Xiaoping, deskilling, discrete time, diversification, diversified portfolio, Doomsday Clock, Edward Thorp, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, eurozone crisis, Everybody Ought to Be Rich, Fall of the Berlin Wall, financial independence, financial innovation, financial thriller, fixed income, full employment, global reserve currency, Goldman Sachs: Vampire Squid, Gordon Gekko, greed is good, happiness index / gross national happiness, haute cuisine, high net worth, Hyman Minsky, index fund, information asymmetry, interest rate swap, invention of the wheel, invisible hand, Isaac Newton, job automation, Johann Wolfgang von Goethe, John Meriwether, joint-stock company, Jones Act, Joseph Schumpeter, Kenneth Arrow, Kenneth Rogoff, Kevin Kelly, laissez-faire capitalism, load shedding, locking in a profit, Long Term Capital Management, Louis Bachelier, margin call, market bubble, market fundamentalism, Marshall McLuhan, Martin Wolf, mega-rich, merger arbitrage, Mikhail Gorbachev, Milgram experiment, money market fund, Mont Pelerin Society, moral hazard, mortgage debt, mortgage tax deduction, mutually assured destruction, Myron Scholes, Naomi Klein, negative equity, NetJets, Network effects, new economy, Nick Leeson, Nixon shock, Northern Rock, nuclear winter, oil shock, Own Your Own Home, Paul Samuelson, pets.com, Philip Mirowski, plutocrats, Plutocrats, Ponzi scheme, price anchoring, price stability, profit maximization, quantitative easing, quantitative trading / quantitative finance, Ralph Nader, RAND corporation, random walk, Ray Kurzweil, regulatory arbitrage, rent control, rent-seeking, reserve currency, Richard Feynman, Richard Thaler, Right to Buy, risk-adjusted returns, risk/return, road to serfdom, Robert Shiller, Robert Shiller, Rod Stewart played at Stephen Schwarzman birthday party, rolodex, Ronald Reagan, Ronald Reagan: Tear down this wall, Satyajit Das, savings glut, shareholder value, Sharpe ratio, short selling, Silicon Valley, six sigma, Slavoj Žižek, South Sea Bubble, special economic zone, statistical model, Stephen Hawking, Steve Jobs, survivorship bias, The Chicago School, The Great Moderation, the market place, the medium is the message, The Myth of the Rational Market, The Nature of the Firm, the new new thing, The Predators' Ball, The Wealth of Nations by Adam Smith, Thorstein Veblen, too big to fail, trickle-down economics, Turing test, Upton Sinclair, value at risk, Yogi Berra, zero-coupon bond, zero-sum game

Random movements in prices, devoid of any trend or cycle, were a depressing prospect for economists. Maurice Kendall, a British statistician, described it as the work of “the Demon of Chance,” randomly drawing a number from a distribution of possible price changes, which, when added to today’s price, determined the next price. While working for a stock market newsletter, Eugene Fama noticed patterns in stock prices that would appear and disappear rapidly. In his doctoral dissertation, he laid out the argument that stock prices were random, reflecting all available information relevant to its value. Prices followed a random walk and market participants could not systematically profit from market inefficiencies. The EMH does not require market price to be always accurate. Investors force the price to fluctuate randomly around its real value.

If you own shares over a year, then most of the time the share price moves up or down a small amount. On some days you may get a large or very large price change. VAR ranks the price changes from largest fall to largest rise. Assuming that prices follow a random walk and price changes fit a normal distribution, you can calculate the probability of a particular size price change. You can answer questions like what is the likely maximum price change and loss on your holding at a specific probability level, say 99 percent, which equates to 1 day out of 100 days. A VAR figure of $50 million at 99 percent over a 10-day holding period means that the bank has a 99 percent probability that it will not suffer a loss of more than $50 million over a 10-day period. VAR became accepted best practice, enshrined in bank regulations. Risk, the unknown unknown, was now a known unknown or even a known known.

On average, investors buying all the stocks in the market would earn higher returns with lower risk. Fund managers with high returns simply took higher risk rather than possessing supernatural skill. Demon of Chance The efficient market hypothesis (EMH) stated that the stock prices followed a random walk, a formal mathematical statement of a trajectory consisting of successive random steps. Pioneers Jules Regnault (in the nineteenth century) and Louis Bachelier (early twentieth century) had discovered that short-term price changes were random—a coin toss could predict up or down moves. Bachelier’s Sorbonne thesis established that the probability of a given change in price was consistent with the Gaussian or bell-shaped normal distribution, well-known in statistical theory. Aware of the importance of his insights, Bachelier claimed: “the present theory resolves the majority of problems in the study of speculation.”5 His examiners disagreed.


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The Success Equation: Untangling Skill and Luck in Business, Sports, and Investing by Michael J. Mauboussin

Amazon Mechanical Turk, Atul Gawande, Benoit Mandelbrot, Black Swan, Checklist Manifesto, Clayton Christensen, cognitive bias, commoditize, Daniel Kahneman / Amos Tversky, David Brooks, deliberate practice, disruptive innovation, Emanuel Derman, fundamental attribution error, Gini coefficient, hindsight bias, hiring and firing, income inequality, Innovator's Dilemma, Long Term Capital Management, loss aversion, Menlo Park, mental accounting, moral hazard, Network effects, prisoner's dilemma, random walk, Richard Thaler, risk-adjusted returns, shareholder value, Simon Singh, six sigma, Steven Pinker, transaction costs, winner-take-all economy, zero-sum game, Zipf's Law

They studied tens of thousands of companies over a forty-year span, amassing over 230,000 firm-years of observations of ROA. These researchers carefully structured the analysis so that it would discriminate between luck and skill in explaining how companies achieved success. The main finding of the study is that “the results consistently indicate that there are many more sustained superior performers than we would expect through the occurrence of lucky random walks.” While this is comforting because it suggests that management's actions, or skill, can lead to success, efforts are ongoing to pinpoint accurately which behaviors were the correct ones. So unlike sports, where there are some observable measures of skill (such as hitting a baseball), all we can really say today is that we cannot explain results by luck alone and that it appears that skill plays a role when companies earn a high return on their assets.

A New Measure That Predicts Performance,” Review of Financial Studies 22, no. 9, September 2009, 3329–3365; and Antti Petajisto, “Active Share and Mutual Fund Performance,” working paper, December 15, 2010. The technical definition of active share: where: ωfund, i = portfolio weight of asset i in the fund ωindex, i = portfolio weight of asset i in the index The sum is taken over the universe of all assets. 27. Jerker Denrell, “Random Walks and Sustained Competitive Advantage,” Management Science 50, no. 7 (July 2004): 922–934. Chapter 8—Building Skill 1. Daniel Kahneman and Gary Klein, “Conditions for Intuitive Expertise: A Failure to Disagree,” American Psychologist 64, no. 6 (September 2009): 515–526. 2. Daniel Kahneman, Thinking, Fast and Slow (New York: Farrar, Straus and Giroux, 2011). 3. Gary Klein, Sources of Power: How People Make Decisions (Cambridge, MA: MIT Press, 1998). 4.

Journal of Financial and Quantitative Analysis 43, no. 4 (December 2008): 907–936. DeLong, J. Bradford, and Kevin Lang. “Are All Economic Hypotheses False?” Journal of Political Economy 100, no. 6 (December 1992): 1257–1272. Denrell, Jerker. “Vicarious Learning, Undersampling of Failure, and the Myths of Management.” Organization Science 14, no. 3 (May–June 2003): 227–243. Denrell, Jerker. “Random Walks and Sustained Competitive Advantage.” Management Science 50, no. 7 (July 2004): 922–934. Derman, Emanuel. Models. Behaving. Badly: Why Confusing Illusion with Reality Can Lead to Disaster, on Wall Street and in Life. New York: Free Press, 2011. Dixon, Mike J., Kevin A. Harrigan, Rajwant Sandhu, Karen Collins, and Jonathan A. Fugelsang. “Losses Disguised as Wins in Modern Multi-line Video Slot Machines.”


pages: 662 words: 180,546

Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown by Philip Mirowski

"Robert Solow", Alvin Roth, Andrei Shleifer, asset-backed security, bank run, barriers to entry, Basel III, Berlin Wall, Bernie Madoff, Bernie Sanders, Black Swan, blue-collar work, Bretton Woods, Brownian motion, business cycle, capital controls, Carmen Reinhart, Cass Sunstein, central bank independence, cognitive dissonance, collapse of Lehman Brothers, collateralized debt obligation, complexity theory, constrained optimization, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, dark matter, David Brooks, David Graeber, debt deflation, deindustrialization, do-ocracy, Edward Glaeser, Eugene Fama: efficient market hypothesis, experimental economics, facts on the ground, Fall of the Berlin Wall, financial deregulation, financial innovation, Flash crash, full employment, George Akerlof, Goldman Sachs: Vampire Squid, Hernando de Soto, housing crisis, Hyman Minsky, illegal immigration, income inequality, incomplete markets, information asymmetry, invisible hand, Jean Tirole, joint-stock company, Kenneth Arrow, Kenneth Rogoff, Kickstarter, knowledge economy, l'esprit de l'escalier, labor-force participation, liberal capitalism, liquidity trap, loose coupling, manufacturing employment, market clearing, market design, market fundamentalism, Martin Wolf, money market fund, Mont Pelerin Society, moral hazard, mortgage debt, Naomi Klein, Nash equilibrium, night-watchman state, Northern Rock, Occupy movement, offshore financial centre, oil shock, Pareto efficiency, Paul Samuelson, payday loans, Philip Mirowski, Ponzi scheme, precariat, prediction markets, price mechanism, profit motive, quantitative easing, race to the bottom, random walk, rent-seeking, Richard Thaler, road to serfdom, Robert Shiller, Robert Shiller, Ronald Coase, Ronald Reagan, savings glut, school choice, sealed-bid auction, Silicon Valley, South Sea Bubble, Steven Levy, technoutopianism, The Chicago School, The Great Moderation, the map is not the territory, The Myth of the Rational Market, the scientific method, The Wisdom of Crowds, theory of mind, Thomas Kuhn: the structure of scientific revolutions, Thorstein Veblen, Tobin tax, too big to fail, transaction costs, Vilfredo Pareto, War on Poverty, Washington Consensus, We are the 99%, working poor

,” on Hyman Minsky influence of on “informational efficacy” and “allocative efficiency,” on Keynesian Theory in New York Review of Books orthodox economics profession on reason for becoming an economist “The Return of Depression Economics,” Kydland–Prescott notion L La Bute, Neil Laibson, David Laissez-faire Lal, Deepak LAMP (Liberal Archief, Ghent) Lanchester, John Landsbanki Lange, Oskar Lasn, Kalle Late Neoliberalism Lehman Brothers Leoni, Bruno Les Mots et les Choses Levin, Richard Levine, David Levitt, Steven Levy, David Lewis, Michael, The Big Short Liberatarianism Liberty Institute Liberty International Liberty League LIBOR scandal Lilly Endowment LinkedIn L’Institut Universitaire des Hautes Etudes Internationales at Geneva Litan, Robert Competitive Equity The Derivatives Dealer’s Club “In Defense of Much, But Not All, Financial Innovation,” writings of Lloyd’s Bank Lo, Andrew on economic crisis Harris & Harris Group Professor of Finance A Non-Random Walk Down Wall Street “Reading About the Financial Crisis,” Lohmann, Larry “Looting: The Economic Underworld of Bankruptcy for Profit” (Romer) Lowenstein, Roger LSE (London School of Economics) Lucas, Robert E. as Bank of Sweden Prize winner on corruption on economic crisis followers of on Keynes neoclassical economists on rational-expectations macroeconomics movement Luntz, Frank M Mack, Christy MacKenzie, Donald MacKinley, A. Craig, A Non-Random Walk Down Wall Street MacroMarkets LLC Madoff, Bernie Make Markets Be Markets (Roosevelt Institute) Mallaby, Sebastian Mankiw, Gregory Marcet, Albert Market Design, Inc.

The journalist Roger Lowenstein declared, “The upside of the current Great Recession is that it could drive a stake through the heart of the academic nostrum known as the efficient-market hypothesis.”57 There was more than sufficient ammunition to choose from to rain fire down on the EMH, not least because it had been the subject of repeated criticism from within the economics profession since the 1980s. But what the journalists like Cassidy, Fox, and Lowenstein and commentators like Krugman neglected to inform their readers was that the back and forth, the intellectual thrust and empirical parry, had ground to a standoff more than a decade before the crisis, as admirably explained in Lo and MacKinlay, A Non-Random Walk Down Wall Street: There is an old joke, widely told among economists, about an economist strolling down the street with a companion when they come upon a $100 bill lying on the ground. As the companion reaches down to pick it up, the economist says, “Don’t bother—if it were a real $100 bill, someone would have already picked it up.” This humorous example of economic logic gone awry strikes dangerously close to home for students of the Efficient Markets Hypothesis, one of the most important controversial and well-studied propositions in all the social sciences.

Competitive Equity: A Better Way to Organize Mutual Funds (Washington: American Enterprise Institute, 2007). Lo, Andrew. “Reading About the Financial Crisis,” Journal of Economic Literature, 50 (2012): 151-178. Lo, Andrew. “Reconciling Efficient Markets with Behavioral Finance: The Adaptive Markets Hypothesis,” Journal of Investment Consulting 7 (2005): 21–44. Lo, Andrew, and Craig MacKinlay. A Non-Random Walk Down Wall Street (Princeton: Princeton University Press, 1999). Loewenstein, George, and Peter Ubel. “Economics Behaving Badly,” New York Times, July 14, 2010. Lofgren, Mike. “Revolt of the Rich,” American Conservative, September 2012. Lohmann, Larry. “Carbon Trading: A Critical Dialogue,” Development Dialogue no. 48, September 2006. Lohmann, Larry. “Carbon Trading, Climate Justice, and the Production of Ignorance,” Development 51 (2008): 359–65.


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The Little Book of Hedge Funds by Anthony Scaramucci

Andrei Shleifer, asset allocation, Bernie Madoff, business process, carried interest, corporate raider, Credit Default Swap, diversification, diversified portfolio, Donald Trump, Eugene Fama: efficient market hypothesis, fear of failure, fixed income, follow your passion, Gordon Gekko, high net worth, index fund, John Meriwether, Long Term Capital Management, mail merge, margin call, mass immigration, merger arbitrage, money market fund, Myron Scholes, NetJets, Ponzi scheme, profit motive, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk-adjusted returns, risk/return, Ronald Reagan, Saturday Night Live, Sharpe ratio, short selling, Silicon Valley, Thales and the olive presses, Thales of Miletus, the new new thing, too big to fail, transaction costs, Vanguard fund, Y2K, Yogi Berra, zero-sum game

Sure, I had heard the term in a Corporate Finance class at Tufts University—my undergraduate alma mater—but the concept barely registered. In plain prose, Professor Malkiel explained that due to perfect information being priced immediately into the markets, the stock prices moved in a random walk. There was no discernible way to predict future prices. Nope. Sorry. No technical analysis, no fundamental analysis, nothing. See, current stock prices were nothing more than a representation of the net present value of the future cash flow streams of each respective company. They were perfectly priced by the market and therefore no one had an edge. If something exogenous happened, well, that would be immediately reflected in price. If you just happened to be on the right side of it, you were the lucky one. You were the monkey on the end of a row of countless monkeys that was flipping a coin and despite the odds it kept coming up heads.

The world of finance was operating under Eugene Fama’s efficient market theory, which was developed in the 1960s at the University of Chicago. Here is the gist of it. If markets were rendered efficient, it followed that prices would move in a random pattern, and consequently those who achieved high levels of success would be investors who most quickly acted upon the fundamental news that was available to everybody. In other words, the only thing that moved a stock price was new information; any other changes were random and not predictable. As such, hedge fund managers did not have an edge . . . or did they? It was April of 1987. I was a first-year law student at Harvard, and desperately wanted to be a summer associate at Goldman Sachs. As I sat in Baker Library, anxiously waiting for my first interview with Goldman Sachs, I picked up A Random Walk on Wall Street by Burton Malkiel. It was then that I got my first exposure to the efficient market theory.

Arbitrage Before we delve into the individual relative value strategies, we must first define arbitrage. Arbitrage is a financial transaction that involves two similar items that are priced differently in different markets. In practice, the trader simultaneously purchases a position in one market and sells the similar position in a different market at a different price. In other words, he is exploiting the price differences of identical positions by buying the same security at a lower price and selling it right away at a higher price. In a perfect scenario, the arbitrageur profits from a difference in the price between the two and earns an immediate profit with no market risk. For example, an announced deal might provide an opportunity for risk arbitrage, or the issuance of a convertible bond by a publicly traded company may signal an opportunity for convertible arbitrage.


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A Devil's Chaplain: Selected Writings by Richard Dawkins

Albert Einstein, Alfred Russel Wallace, Buckminster Fuller, butterfly effect, Claude Shannon: information theory, complexity theory, Desert Island Discs, double helix, Douglas Hofstadter, epigenetics, experimental subject, Fellow of the Royal Society, gravity well, Necker cube, out of africa, phenotype, placebo effect, random walk, Richard Feynman, Silicon Valley, stem cell, Stephen Hawking, the scientific method

A belief in the ubiquity of gradualistic evolution does not necessarily commit us to Darwinian natural selection as the steering mechanism guiding the search through genetic space. It is highly probable that Motoo Kimura is right to insist that most of the evolutionary steps taken through genetic space are unsteered steps. To a large extent the trajectory of small, gradualistic steps actually taken may constitute a random walk rather than a walk guided by selection. But this is irrelevant if – for the reasons given above – our concern is with adaptive evolution as opposed to evolutionary change per se. Kimura himself rightly insists9 that his ‘neutral theory is not antagonistic to the cherished view that evolution of form and function is guided by Darwinian selection’. Further, the theory does not deny the role of natural selection in determining the course of adaptive evolution, but it assumes that only a minute fraction of DNA changes in evolution are adaptive in nature, while the great majority of phenotypically silent molecular substitutions exert no significant influence on survival and reproduction and drift randomly through the species.

Further, the theory does not deny the role of natural selection in determining the course of adaptive evolution, but it assumes that only a minute fraction of DNA changes in evolution are adaptive in nature, while the great majority of phenotypically silent molecular substitutions exert no significant influence on survival and reproduction and drift randomly through the species. The facts of adaptation compel us to the conclusion that evolutionary trajectories are not all random. There has to be some nonrandom guidance towards adaptive solutions because nonrandom is what adaptive solutions precisely are. Neither random walk nor random saltation can do the trick on its own. But does the guiding mechanism necessarily have to be the Darwinian one of nonrandom survival of random spontaneous variation? The obvious alternative class of theory postulates some form of nonrandom, i.e. directed, variation. Nonrandom, in this context, means directed towards adaptation. It does not mean causeless. Mutations are, of course, caused by physical events, for instance, cosmic ray bombardment.

Whereas I would do so on logical grounds, Gould prefers an empirical assault. He looks at the actual course of evolution and argues that such apparent progress as can in general be detected is artefactual (like the baseball statistic). Cope’s rule of increased body size, for example, follows from a simple ‘drunkard’s walk’ model. The distribution of possible sizes is confined by a left wall, a minimal size. A random walk from a beginning near the left wall has nowhere to go but up the size distribution. The mean size has pretty well got to increase, and it doesn’t imply a driven evolutionary trend towards larger size. As Gould convincingly argues, the effect is compounded by a human tendency to give undue weight to new arrivals on the geological scene. Textbook biological histories emphasize a progression of grades of organization.


Data Mining the Web: Uncovering Patterns in Web Content, Structure, and Usage by Zdravko Markov, Daniel T. Larose

Firefox, information retrieval, Internet Archive, iterative process, natural language processing, pattern recognition, random walk, recommendation engine, semantic web, speech recognition, statistical model, William of Occam

c 0.726 b 0.413 50 CHAPTER 2 HYPERLINK-BASED RANKING PAGERANK The hyperlinks are not only ways to propagate the prestige score of a page to pages to which it links, they are also paths along which web users travel from one web page to another. In this respect, the popularity (or prestige) of a web page can be measured in terms of how often an average web user visits it. To estimate this we may use the metaphor of the “random web surfer,” who clicks on hyperlinks at random with uniform probability and thus implements the random walk on the web graph. Assume that page u links to Nu web pages and page v is one of them. Then once the web surfer is at page u, the probability of visiting page v will be 1/Nu . This intuition suggests a more sophisticated scheme of propagation of prestige through the web links also involving the out-degree of the nodes. The idea is that the amount of prestige that page v receives from page u is 1/Nu of the prestige of u.

The simplified PageRank algorithm generally works with loops; however, there is a special configuration of nodes that the algorithm cannot deal with properly. Consider, for example, two pages that point to each other but do not point to other pages. Such an isolated loop is called a rank sink. If pointed to from an outside page, it accumulates rank but never distributes it to other nodes. To deal with the rank sink situation, we return to the random surfer model. As we have already noted, computing page rank is based on the idea of a random walk on the web graph, but the random surfer may get trapped into a rank sink. To avoid this situation we try to model the behavior of a real web surfer who gets bored running into a loop and jumps to some other web page outside the rank sink. For this purpose we introduce a rank source E, a vector over all web pages, which defines the probability distribution of jumping to a web page at random. Thus, the modified PageRank equation becomes A(v, u)R(v) R(u) = λ + E(u) Nv v The PageRank equation can be solved by using the eigenvector approach.

Below we present an iterative algorithm, which basically implements the power iteration 52 CHAPTER 2 HYPERLINK-BASED RANKING method for computing the dominant eigenvector with a small modification of the way the normalization is done. r R ← R0 r loop: ◦ Q←R R ← A TQ ◦ d ← Q1 − R1 ◦ ◦ R ← R + dE r while R − Q > ε 1 The initial rank vector R0 can be any vector over the web pages, A T is the transpose of the adjacency matrix with weights 1/Nu , and E is the rank source vector. The parameter d implements the normalization step and also affects the rate of convergence positively. The alternative approach would be just to add E to R and then normalize (R ← R/R1 ). As defined, the PageRank algorithm implements the random surfer model, where: r The rank vector R defines the probability distribution of a random walk on the graph of the Web. r With some low probability the surfer jumps to a random page chosen according to the distribution E. The source of rank E is usually chosen as a uniform vector with a small norm (e.g., E1 = 0.15). The way it affects the model of the random surfer is that the jumps to a random page happen more often if the norm of E is larger. In terms of PageRank score, a larger E means less contribution of the link structure to the final score (i.e., the rank distribution in R gets closer to E).


pages: 443 words: 51,804

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

algorithmic trading, asset allocation, automated trading system, backtesting, Black-Scholes formula, Brownian motion, business process, buy and hold, continuous integration, corporate governance, discrete time, distributed generation, fixed income, Flash crash, housing crisis, implied volatility, incomplete markets, linear programming, mandelbrot fractal, market friction, market microstructure, martingale, Menlo Park, p-value, pattern recognition, performance metric, principal–agent problem, random walk, risk tolerance, risk/return, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process

See also Volatility index (VIX) pVIX cVIX spread, 106 Qiu, Hongwei, xiv, 97 Q-learning algorithm, 65 Quadratic covariation formula, 244 Quadratic covariation-realized covariance estimator, 266 Quadratic utility function, 286 Quadratic variation, estimate of, 224 Quadrinomial tree method, 99–100 volatility index convergence and, 105 vs. CBOE procedure, 100–101 Quantile–quantile (QQ) plots, 80 of empirical CDF, 136 of high-frequency tranche prices, 92, 94 of tranche prices, 83–84 ‘‘Quantile type’’ rule, 30 Quantum mechanics, 385 Quote-to-quote returns, 258, 260 Random variables, 334–336 Random walk, 126 Rare-event analysis, 32–33 Rare-event detection, 28, 30–32 Rare events detecting and evaluating, 29–35 equity price and, 44 trades profile and, 42, 43 Rare-events distribution, 41–44 peaks in, 42 Real daily integrated covariance, regressing, 281 Real integrated covariance regressions, results of, 282–285 Realized covariance (RC), 269 estimator for, 280 measures of, 272 Realized covariance plus leads and lags (RCLL), 266 estimator for, 280, 290 Realized covariance–quadratic variation estimator, 244 Realized variance, 12 Realized volatility, microstructure noise and, 274 Index Realized volatility estimator, 253–254, 256 results of, 276–279 Realized volatility estimator performance, ranking, 279 Realized-volatility-type measures, 275 Real-valued functions, 350, 351, 388–389 Refresh time, 267 Refresh time procedure, 244 Refresh time synchronization method, 268 Regime-switching default correlation, 81–84 Regime-switching default correlation model, 76 Regime-switching model, drawback of, 84–85 ‘‘Regret-free’’ prices, 238 Regular asynchronous trading, 264 Regular nonsynchronous trading, 268 Regular synchronous trading, 268 Relative risk process, 296 Rellich’s theorem, 398 Representative ADT algorithm, 52–53, 54.

Additionally, the combination of Adaboost and the BSC can be used as a semiautomated strategic planning system that continuously updates itself for board-level decisions of directors or for investment decisions of portfolio managers. REFERENCES Acharya VV, John K, Sundaram RK. On the optimality of resetting executive stock options. J Financ Econ 2000;57:65–101. Alexander S. Price movements in speculative markets: trends or random walks. Ind Manag Rev 1961;2:7–26. Algoet PH, Cover TM. Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann Probab 1988;16:876–898. Allen F, Karjalainen R. Using genetic algorithms to find technical trading rules. J Financ Econ 1999;51:245–271. Altman EI. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy.

See also Constant rebalanced portfolio technical analysis 436 Portfolios (continued ) (CRP-TA) trading algorithm; Multiagent portfolio management system; Subprime MBS portfolios MBS, 77 tranches of, 77 vintage of, 77 Portfolios value, expected change in, 385 Portfolio utility, 286 Position strategy, 33 Positive process, 310 Powell’s method, 6, 14, 19 Power-type utility functions, 305 Preaveraging technique, 267 Prediction nodes, 50, 51 Prediction rule, 48, 49 Prespecified terminal time, 295 Price behavior, analyzing after rare events, 28 Price change distributions, 31 Price distribution distortion, 91 Price evolution in time, 30 Price movement(s) corresponding to small volume, 30 detecting and evaluating, 44 persistence of, 27–46 Price movement methodology, results of, 35–41 Price process, 121 Price recovery probability of, 44 after rare events, 45 Price volatility, UHFT and, 241 Price–volume relationship, 27–28 outlying observations of, 28 Principal–agent conflict, 53 Principal–agent problem, 60 Probability of favorable price movement, 35–36 Poisson, 240 Probability density, 13–14 Probability density function (pdf), 119, 120, 163, 171, 335. See also Forecast pdfs; pdf forecasting; Sample pdfs Probability distributions, 165 Probability mass function (pmf), 171 Probability surfaces, 35, 37 Proportionality constant, 402 Pure optimal stopping problems, 311 Put options, demand for, 106 Index Put options chains, constructed VIX using, 105–106 p-values, 138–139, 204–205 pVIX-b, 102–103, 105.


pages: 111 words: 1

Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb

Antoine Gombaud: Chevalier de Méré, availability heuristic, backtesting, Benoit Mandelbrot, Black Swan, commoditize, complexity theory, corporate governance, corporate raider, currency peg, Daniel Kahneman / Amos Tversky, discounted cash flows, diversified portfolio, endowment effect, equity premium, fixed income, global village, hedonic treadmill, hindsight bias, Kenneth Arrow, Long Term Capital Management, loss aversion, mandelbrot fractal, mental accounting, meta analysis, meta-analysis, Myron Scholes, Paul Samuelson, quantitative trading / quantitative finance, QWERTY keyboard, random walk, Richard Feynman, road to serfdom, Robert Shiller, Robert Shiller, selection bias, shareholder value, Sharpe ratio, Steven Pinker, stochastic process, survivorship bias, too big to fail, Turing test, Yogi Berra

With no mathematical literacy we can launch a Monte Carlo simulation of an eighteen-year-old Christian Lebanese successively playing Russian roulette for a given sum, and see how many of these attempts result in enrichment, or how long it takes on average before he hits the obituary. We can change the barrel to contain 500 holes, a matter that would decrease the probability of death, and see the results. Monte Carlo simulation methods were pioneered in martial physics in the Los Alamos laboratory during the A-bomb preparation. They became popular in financial mathematics in the 1980s, particularly in the theories of the random walk of asset prices. Clearly, we have to say that the example of Russian roulette does not need such apparatus, but many problems, particularly those resembling real-life situations, require the potency of a Monte Carlo simulator. Monte Carlo Mathematics It is a fact that “true” mathematicians do not like Monte Carlo methods. They believe that they rob us of the finesse and elegance of mathematics. They call it “brute force.”

Literally every great thinker has dabbled with it, most of them obsessively. The two greatest minds to me, Einstein and Keynes, both started their intellectual journeys with it. Einstein wrote a major paper in 1905, in which he was almost the first to examine in probabilistic terms the succession of random events, namely the evolution of suspended particles in a stationary liquid. His article on the theory of the Brownian movement can be used as the backbone of the random walk approach used in financial modeling. As for Keynes, to the literate person he is not the political economist that tweed-clad leftists love to quote, but the author of the magisterial, introspective, and potent Treatise on Probability. For before his venturing into the murky field of political economy, Keynes was a probabilist. He also had other interesting attributes (he blew up trading his account after experiencing excessive opulence—people’s understanding of probability does not translate into their behavior).

While early economic models excluded randomness, Arthur explained how “unexpected orders, chance meetings with lawyers, managerial whims . . . would help determine which ones achieved early sales and, over time, which firms dominated.” MATHEMATICS INSIDE AND OUTSIDE THE REAL WORLD A mathematical approach to the problem is in order. While in conventional models (such as the well-known Brownian random walk used in finance) the probability of success does not change with every incremental step, only the accumulated wealth, Arthur suggests models such as the Polya process, which is mathematically very difficult to work with, but can be easily understood with the aid of a Monte Carlo simulator. The Polya process can be presented as follows: Assume an urn initially containing equal quantities of black and red balls.


pages: 543 words: 147,357

Them And Us: Politics, Greed And Inequality - Why We Need A Fair Society by Will Hutton

Andrei Shleifer, asset-backed security, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Big bang: deregulation of the City of London, Blythe Masters, Boris Johnson, Bretton Woods, business cycle, capital controls, carbon footprint, Carmen Reinhart, Cass Sunstein, centre right, choice architecture, cloud computing, collective bargaining, conceptual framework, Corn Laws, corporate governance, creative destruction, credit crunch, Credit Default Swap, debt deflation, decarbonisation, Deng Xiaoping, discovery of DNA, discovery of the americas, discrete time, diversification, double helix, Edward Glaeser, financial deregulation, financial innovation, financial intermediation, first-past-the-post, floating exchange rates, Francis Fukuyama: the end of history, Frank Levy and Richard Murnane: The New Division of Labor, full employment, George Akerlof, Gini coefficient, global supply chain, Growth in a Time of Debt, Hyman Minsky, I think there is a world market for maybe five computers, income inequality, inflation targeting, interest rate swap, invisible hand, Isaac Newton, James Dyson, James Watt: steam engine, joint-stock company, Joseph Schumpeter, Kenneth Rogoff, knowledge economy, knowledge worker, labour market flexibility, liberal capitalism, light touch regulation, Long Term Capital Management, Louis Pasteur, low cost airline, low-wage service sector, mandelbrot fractal, margin call, market fundamentalism, Martin Wolf, mass immigration, means of production, Mikhail Gorbachev, millennium bug, money market fund, moral hazard, moral panic, mortgage debt, Myron Scholes, Neil Kinnock, new economy, Northern Rock, offshore financial centre, open economy, plutocrats, Plutocrats, price discrimination, private sector deleveraging, purchasing power parity, quantitative easing, race to the bottom, railway mania, random walk, rent-seeking, reserve currency, Richard Thaler, Right to Buy, rising living standards, Robert Shiller, Robert Shiller, Ronald Reagan, Rory Sutherland, Satyajit Das, shareholder value, short selling, Silicon Valley, Skype, South Sea Bubble, Steve Jobs, The Market for Lemons, the market place, The Myth of the Rational Market, the payments system, the scientific method, The Wealth of Nations by Adam Smith, too big to fail, unpaid internship, value at risk, Vilfredo Pareto, Washington Consensus, wealth creators, working poor, zero-sum game, éminence grise

Happily ignoring the accumulated wisdom of Russell, Knight, Keynes and Newton, from the 1960s onwards, a group of mathematical economists hypothesised that the financial markets were different. There is abundant data about the movement of the prices of financial assets, although actually defining the universe of data proved much more problematic in practice. If you make the assumptions that financial markets are efficient containing all the information that they can, and that consequently all price movements are independent of each other and cannot be related to each other or the past, then important conclusions follow. Financial prices will move wholly randomly, as likely to go up as down – the ‘random walk’. If this is true then, as mentioned earlier, financial data will correspond to the law of large numbers and follow the same rules that dictate the distribution of, say, tall, average and short people, dice rolls and flips of a coin.

It could only be a legitimate question if the markets were not efficient, prices were not randomly distributed and events were not distributed on Gaussian principles, but nobody who wanted to stay in the mainstream could suggest such things. There is an enormous intellectual and financial investment in the status quo. Academics have built careers, reputations and tenure on a particular view of the world being right. Only an earthquake can persuade them to put up their hands and acknowledge they were wrong. When the mathematician Benoit Mandelbrot began developing his so-called fractal mathematics and power laws in the early 1960s, arguing that the big events outside the normal distribution are the ones that need explaining and assaulting the whole edifice of mathematical theory and the random walk, MIT’s Professor Paul Cootner (the great random walk theorist) exclaimed: ‘surely, before consigning centuries of work to the ash pile, we should like some assurance that all our work is truly useless’.

What Weatherstone wanted to know was how much money the bank would lose if it were hit by a big event outside the normal distribution of events. Such events are statistically improbable but still possible. But would they present too much risk, and bring down the whole bank? This led to the development of mathematically computed value at risk (VaR), which was based on the same assumptions about random walks, efficient markets and bell curves that had been used when pricing derivatives. The VaR figure is the maximum amount a financial institution might lose on any given day with a probability of 95 per cent or higher. Dick Fuld, the CEO of Lehman Brothers, could comfort himself throughout 2007 and even the first half of 2008 that his bank was exposed to less than $100 million of VaR on any given day (between 95 and 99 per cent confidence level).


pages: 339 words: 94,769

Possible Minds: Twenty-Five Ways of Looking at AI by John Brockman

AI winter, airport security, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, artificial general intelligence, Asilomar, autonomous vehicles, basic income, Benoit Mandelbrot, Bill Joy: nanobots, Buckminster Fuller, cellular automata, Claude Shannon: information theory, Daniel Kahneman / Amos Tversky, Danny Hillis, David Graeber, easy for humans, difficult for computers, Elon Musk, Eratosthenes, Ernest Rutherford, finite state, friendly AI, future of work, Geoffrey West, Santa Fe Institute, gig economy, income inequality, industrial robot, information retrieval, invention of writing, James Watt: steam engine, Johannes Kepler, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Maynard Keynes: technological unemployment, John von Neumann, Kevin Kelly, Kickstarter, Laplace demon, Loebner Prize, market fundamentalism, Marshall McLuhan, Menlo Park, Norbert Wiener, optical character recognition, pattern recognition, personalized medicine, Picturephone, profit maximization, profit motive, RAND corporation, random walk, Ray Kurzweil, Richard Feynman, Rodney Brooks, self-driving car, sexual politics, Silicon Valley, Skype, social graph, speech recognition, statistical model, Stephen Hawking, Steven Pinker, Stewart Brand, strong AI, superintelligent machines, supervolcano, technological singularity, technoutopianism, telemarketer, telerobotics, the scientific method, theory of mind, Turing machine, Turing test, universal basic income, Upton Sinclair, Von Neumann architecture, Whole Earth Catalog, Y2K, zero-sum game

If you are walking in a mountainous region and want to get home, always walking downhill will most likely get you to the next valley but not necessarily over the other mountains that lie around it and between you and home. For that, you either need a mental model (i.e., a map) of the topology, so you know where to ascend to get out of the valley, or you need to switch between gradient descent and random walks so you can bounce your way out of the region. Which is, in fact, exactly what the mosquito does in following my scent: It descends when it’s in my plume and random-walks when it has lost the trail or hit an obstacle. AI So that’s nature. What about computers? Traditional software doesn’t work that way—it follows deterministic trees of hard logic: “If this, do that.” But software that interacts with the physical world tends to work more like the physical world. That means dealing with noisy inputs (sensors or human behavior) and providing probabilistic, not deterministic, results.

What seems like the powerful radar of insects in the dark, with blood-seeking intelligence inexplicable for such tiny brains, is actually just a sensitive nose with almost no intelligence at all. Mosquitoes are closer to plants that follow the sun than to guided missiles. Yet by applying this simple “follow your nose” rule quite literally, they can travel through a house to find you, slip through cracks in a screen door, even zero in on the tiny strip of skin you left exposed between hat and shirt collar. It’s just a random walk, combined with flexible wings and legs that let the insect bounce off obstacles and an instinct to descend a chemical gradient. But “gradient descent” is much more than bug navigation. Look around you and you’ll find it everywhere, from the most basic physical rules of the universe to the most advanced artificial intelligence. THE UNIVERSE We live in a world of countless gradients, from light and heat to gravity and chemical trails (chemtrails!).

By dint of everyday experience, we have grown used to the fact that airport security is different for children under the age of twelve and adults over the age of seventy-five. What factors do we want the algorists to have in their often hidden procedures? Education? Income? Employment history? What one has read, watched, visited, or bought? Prior contact with law enforcement? How do we want algorists to weight those factors? Predictive analytics predicated on mechanical objectivity comes at a price. Sometimes it may be a price worth paying; sometimes that price would be devastating for the just society we want to have. More generally, as the convergence of algorithms and Big Data governs a greater and greater part of our lives, it would be well worth keeping in mind these two lessons from the history of the sciences: Judgment is not the discarded husk of a now pure objectivity of self-restraint. And mechanical objectivity is a virtue competing among others, not the defining essence of the scientific enterprise.


pages: 405 words: 109,114

Unfinished Business by Tamim Bayoumi

algorithmic trading, Asian financial crisis, bank run, banking crisis, Basel III, battle of ideas, Ben Bernanke: helicopter money, Berlin Wall, Big bang: deregulation of the City of London, Bretton Woods, British Empire, business cycle, buy and hold, capital controls, Celtic Tiger, central bank independence, collapse of Lehman Brothers, collateralized debt obligation, credit crunch, currency manipulation / currency intervention, currency peg, Doha Development Round, facts on the ground, Fall of the Berlin Wall, financial deregulation, floating exchange rates, full employment, hiring and firing, housing crisis, inflation targeting, Just-in-time delivery, Kenneth Rogoff, liberal capitalism, light touch regulation, London Interbank Offered Rate, Long Term Capital Management, market bubble, Martin Wolf, moral hazard, oil shale / tar sands, oil shock, price stability, prisoner's dilemma, profit maximization, quantitative easing, race to the bottom, random walk, reserve currency, Robert Shiller, Robert Shiller, Rubik’s Cube, savings glut, technology bubble, The Great Moderation, The Myth of the Rational Market, the payments system, The Wisdom of Crowds, too big to fail, trade liberalization, transaction costs, value at risk

Clearly, if General Motors announces profits that are higher than investors expect then the price of its shares will increase. However, it is equally likely that the profit announcement will disappoint and shares will go down. Hence, before the announcement, the direction of the share price is unpredictable—it is as likely to rise as to fall. This is often called the random walk theory, as the same properties are exhibited by a random walk in which, while taking a step forward, a person—usually assumed to be drunk—is as likely to also lurch to the left as to the right. Because the left-right movement is unpredictable, the best guess as to where the drunk will be at the next step is straight ahead. While there are some modest deviations from the prediction that markets are as likely to surprise on the upside as on the downside, as a whole it stands up pretty well.

The booms primarily involved excessive borrowing, much of which was used to buy houses. The resulting increase in house prices, however, had little impact on the measures of consumer price inflation that the central banks focused on. In the United States, the consumer price index used rents rather than new house prices to estimate the cost of housing as this is (correctly) seen as more direct measure of the price of shelter. As rents did not take off in the same way that house prices did, the impact of the housing bubble on consumer price inflation was muted. In the European Union, house prices had no direct impact on consumer price inflation as, in the absence of a uniform way of measuring dwelling costs across the member countries, the consumer price index used by the ECB excluded any measure of housing costs. In addition, much of the additional spending on goods was satisfied by higher imports.

., (i), (ii) North Atlantic crisis and Basel rules, (i) causes, (i) and currency unions, (i) and debt flows, (i) and economic models, (i) effects and consequences, (i), (ii), (iii), (iv) ends European banking boom, (i) European monetary union effect on, (i) and fall in confidence in experts, (i) financial boom and bust, (i), (ii) misery index, (i) origins (August 2007), (i), (ii), (iii) output losses, (i), (ii) responses to, (i), (ii), (iii), (iv) responsibility for, (i) and speculative ventures, (i) unpreparedness, (i), (ii), (iii) as watershed event, (i) Norway invited to join European Economic Community, (i) in Scandinavian monetary union, (i) Obama, Barack, (i) Office of the Comptroller of the Currency (OCC; US), (i), (ii), (iii) oil prices, (i), (ii) Organisation for Economic Cooperation and Development (OECD), (i) output losses, (i), (ii) volatility, (i) Outright Monetary Transaction (OMT, Euro area), (i), (ii), (iii) Padoa-Schioppa, Tommaso, (i), (ii) Parvest Dynamic ABS, (i) petrodollars, (i), (ii) Philippines, (i) physics: parallel with economic models, (i) Plaza Agreement (1985), (i), (ii), (iii) Pöhl, Karl Otto, (i), (ii), (iii) Pompidou, Georges, (i) Portugal borrowing interest rate, (i) commercial loans, (i) connected firms in, (i) in currency union periphery, (i) in Euro area, (i) European aid to, (i) excessive borrowing, (i) in Exchange Rate Mechanism, (i) expansion in bank assets, (i) financial crisis in, (i), (ii), (iii) high interest rates, (i) product market improvements, (i) reduces fiscal deficit, (i) ten-year bonds, (i) pound sterling (UK currency) Bundesbank ceases to support, (i), (ii) devalued, (i) diminishing role, (i) leaves ERM, (i) prisoners’ dilemma, (i) public sector borrowers, (i) quantitative easing, (i) quantum mechanics, parallels to economics, (i), (ii) Quantitative Impact Studies (QISs), (i) Rajan, Raghuram, (i) random walk theory, (i) rational expectations, (i) Reagan, Ronald, (i), (ii), (iii), (iv) Regulation Q see United States of America Reigal Neal Interstate Branching Efficiency Act (US, 1997), (i) renmimbi (Chinese currency): depreciation against dollar (1994), (i) repurchase agreements (repos) broker-dealers exploit, (i) collateral, (i), (ii), (iii), (iv) expansion, (i) and foreign borrowing, (i) freeze, (i) fund housing bubble, (i) liquidity, (i) market shrinks in US, (i) as source of investment bank funding, (i) Ricardian equivalence, (i) Ricardian offset, (i) risk models, (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv) risk-weighted assets, (i), (ii), (iii) Rochard, Michel, (i) Rome, Treaty of (1957), (i), (ii), (iii), (iv), (v), (vi) Roosevelt, Franklin D., (i) Roubini, Nouriel, (i) Royal Bank of Scotland (RBS; UK bank), (i) Russia exchange rate collapse (1998), (i) joins WTO, (i) safe haven bankruptcy protection (US), (i) Sanio, Jochen, (i) Santander (Spanish bank) assets expanded, (i) capitalization, (i) international scope, (i) as mega-bank, (i) takeovers, (i) Sants, Hector, (i) Savings and Loans (US), (i), (ii) Scandinavia: monetary union, (i) Schmidt, Helmut, (i), (ii), (iii) Schoales, Myron, (i) Securities and Exchange Commission (SEC; US) and mortgage-backed securities, (i) registers hedge funds, (i) and regulation, (i), (ii), (iii), (iv) Release 47683 widens repurchase agreement collateral, (i) and repo market, (i), (ii), (iii), (iv) securitization and mortgages, (i), (ii), (iii), (iv), (v), (vi) private label, (i) in US, (i), (ii), (iii), (iv), (v), (vi) Security Pacific Corporation (US bank), (i) shadow banks see United States of America share prices: fluctuations and predictions, (i), (ii) Shiller, Robert, (i), (ii), (iii) Silva-Herzog, Jesus, (i) silver: in US money supply, (i) single currency benefits, (i) effect on trade, (i) and macroeconomic shocks, (i) see also Euro area Single European Act (1986), (i), (ii), (iii), (iv), (v) Smith, Adam, (i), (ii) Smithsonian Agreement, (i), (ii) snake currency arrangement (Europe), (i), (ii) Société Générale (French bank): expansion, (i), (ii), (iii) South Korea, (i), (ii), (iii), (iv) Spain borrowing interest rate, (i) caja savings banks, (i) commercial loans, (i) in currency union periphery, (i) ejected from Exchange Rate Mechanism, (i) excessive borrowing, (i) in Exchange Rate Mechanism, (i) expansion in bank assets, (i) and ESM funding to restructure banking system, (i) financial crisis in, (i), (ii), (iii) high interest rates, (i) housing, (i), (ii) included in Euro area, (i) local governments in, (i) reduces fiscal deficit, (i) successful effect of reforms, (i) ten-year bonds, (i) Stability and Growth Pact (SGP, Euro area), (i), (ii), (iii), (iv), (v) Strasbourg summit (of European leaders, 1990), (i), (ii) Strauss-Kahn, Dominique, (i) stressed value-at-risk models (SVARs), (i) Suez (French bank), (i) supply and demand, law of, (i) Sweden in Basel Committee, (i) and currency fluctuations, (i) in Scandinavian monetary union, (i) Switzerland in financial crisis, (i) trade with EMU members, (i) taxes cuts, (i), (ii), (iii), (iv) favor debt over equity, (i) kept low, (i) little effect on private spending, (i) TCW (US bank), (i) Texas: house prices fall, (i) Thailand, (i), (ii), (iii), (iv) Thatcher, Margaret, (i), (ii), (iii), (iv), (v), (vi) trade: affected by single currency, (i) trade balance, (i) Travelers Group (US financial institution), (i) Trichet, Jean-Claude, (i) Trump, Donald elected President, (i) and fiscal stimulus, (i) looser view on bank regulation, (i) proposes tax cuts, (i) UBS (Swiss bank), (i) UniCredit (Italian bank), (i), (ii), (iii) United Kingdom (Britain) bank assets reduced since 2008, (i) banking expansion, (i), (ii) banking system, (i) bond markets, (i) central bank independence, (i) common capital standard agreed with US, (i) core Euro banks expand into, (i) and currency fluctuations, (i) favors larger bank capital buffers, (i) favours EU-wide bank regulator, (i) financial crisis (1866), (i) foreign banks in, (i) foreign investments in, (i) high inflation, (i), (ii) invited to join European Economic Community, (i) joins Exchange Rate Mechanism, (i) large outflows, (i) leaves European Union, (i) leaves Exchange Rate Mechanism, (i), (ii) ‘light touch’ regulation, (i), (ii), (iii), (iv), (v) in North Atlantic financial crisis (2008), (i) opts out of Maastricht Treaty, (i) owns US assets, (i), (ii) product market, (i) rebate from EEC budget, (i) resists monetary union, (i), (ii) scale of banking, (i) separated commercial and investment (merchant) banks, (i) trade with EMU members, (i) see also pound sterling United States of America accepts Basel 3 framework for large banks, (i) accounting practices, (i) adopts new leverage ratio, (i) aggregate spending, (i) anchor regions and peripheries in currency union, (i) and Asian crisis, (i) assets held by European banks, (i), (ii) bank assets reduced since 2008, (i) bank deposits migrate, (i) bank failures and prompt corrective action, (i) bank mergers, (i) bank size compared with Europe, (i) bankers’ morality, (i) banking expansion, (i) banking regulation, (i), (ii), (iii), (iv), (v), (vi), (vii) and Basel 2 accord, (i) in Basel Committee, (i) bond markets, (i) bond yields fall, (i) business cycles, (i) champions internal risk models, (i) common capital standard agreed with UK, (i) consumer price index, (i) core Euro banks expand into, (i) as crisis country, (i) currency as international standard, (i) currency union in, (i), (ii), (iii) debt outflows, (i) deregulation, (i), (ii) devaluation, (i) effect of break-up of Bretton Woods on, (i) effect of post-crisis changes, (i), (ii) Euro area lends to, (i), (ii) European universal banks in, (i), (ii) favors larger bank capital buffers, (i) federal support for banks, (i) federal tax system, (i) financial boom, (i), (ii) financial reform in, (i), (ii) financial system (2002), (i), (ii) floating exchange rates, (i) Flow of Funds data, (i), (ii) fractured banking system, (i) and gold market, (i) high tech boom collapses, (i) house prices, (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv) imposes surcharge on foreign imports, (i) improved monetary policy, (i) inflation fluctuates, (i), (ii), (iii), (iv), (v), (vi) integrated banking system, (i) interest rates limited, (i), (ii), (iii) investment bank expansion, (i), (ii) investment bank regulation, (i), (ii), (iii) labor market flexibility, (i) and Latin American debt crisis, (i), (ii) misery index, (i) modest recovery from crisis, (i) national (interstate) banks, (i), (ii), (iii), (iv) national price movements, (i) and oil prices, (i) output volatility, (i) policy coordination fades, (i) post 2002 financial boom, (i) product market, (i) recessions (1985–2005), (i) regulation of shadow banks, (i) and repo market, (i) response to crisis, (i) responsibility for macroprudential policies, (i) and risk measures, (i), (ii) securitization, (i), (ii) separates commercial and investment banking, (i), (ii), (iii) shadow banks develop, (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x) and small bank regulation, (i) trade balance, (i), (ii) trade with EMU members, (i) unprepared for financial crisis, (i) United States Federal Reserve Bank belief in market discipline of investment banks, (i), (ii), (iii), (iv), (v) and business cycle, (i) conducts stress tests, (i) cooperation of monetary and fiscal policy, (i) eases rates, (i), (ii) easy financing conditions, (i) emergency funding, (i), (ii) faith in investors’ judgment, (i) helps stabilize markets, (i) and house price boom, (i) and inflation rates, (i) monetary policy, (i) as proposed model for European Central Bank, (i) provides safety net, (i), (ii) regulates mortgage lending standards, (i) and regulation of investment banks, (i), (ii) Regulation Q, (i) regulatory function and practice, (i), (ii), (iii), (iv) response to crisis, (i) and risk models, (i), (ii), (iii) and tax cuts, (i) urges reform of Basel (i), (ii) warns about loans to Latin America, (i) value-at-risk models (VARs), (i) Venezuela, (i) Versailles Treaty (1919), (i) Vietnam War, (i) Volcker, Paul, (i), (ii) rule, (i) Wachovia Corporation (US bank), (i) Wall Street: reform, (i) waterfall investment structures, (i) welfare payments, (i) Wells Fargo (US bank), (i), (ii) Werner Commission Report (Europe) (1970), (i), (ii), (iii) West Germany economic growth, (i) in European Coal and Steel Community, (i) see also Germany White, Bill, (i) White, Harry Dexter, (i) won (S.


pages: 312 words: 35,664

The Mathematics of Banking and Finance by Dennis W. Cox, Michael A. A. Cox

barriers to entry, Brownian motion, call centre, correlation coefficient, fixed income, G4S, inventory management, iterative process, linear programming, meta analysis, meta-analysis, pattern recognition, random walk, traveling salesman, value at risk

Further, a 2 is equally likely to appear on any subsequent roll of the dice. 4.3 ESTIMATION OF PROBABILITIES There are a number of ways in which you can arrive at an estimate of Prob(A) for the event A. Three possible approaches are: r A subjective approach, or ‘guess work’, which is used when an experiment cannot be easily r repeated, even conceptually. Typical examples of this include horse racing and Brownian motion. Brownian motion represents the random motion of small particles suspended in a gas or liquid and is seen, for example, in the random walk pattern of a drunken man. The classical approach, which is usually adopted if all sample points are equally likely (as is the case in the rolling of a dice as discussed above). The probability may be measured with certainty by analysing the event. Using the same mathematical notation, a mathematical definition of this is: Prob(A) = Number of events classifiable as A Total number of possible events A typical example of such a probability is a lottery.

Index a notation 103–4, 107–20, 135–47 linear regression 103–4, 107–20 slope significance test 112–20 variance 112 abscissa see horizontal axis absolute value, notation 282–4 accuracy and reliability, data 17, 47 adaptive resonance theory 275 addition, mathematical notation 279 addition of normal variables, normal distribution 70 addition rule, probability theory 24–5 additional variables, linear programming 167–70 adjusted cash flows, concepts 228–9 adjusted discount rates, concepts 228–9 Advanced Measurement Approach (AMA) 271 advertising allocation, linear programming 154–7 air-conditioning units 182–5 algorithms, neural networks 275–6 alternatives, decisions 191–4 AMA see Advanced Measurement Approach analysis data 47–52, 129–47, 271–4 Latin squares 131–2, 143–7 linear regression 110–20 projects 190–2, 219–25, 228–34 randomised block design 129–35 sampling 47–52, 129–47 scenario analysis 40, 193–4, 271–4 trends 235–47 two-way classification 135–47 variance 110–20, 121–7 anonimised databases, scenario analysis