Paul Lévy

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pages: 364 words: 101,286

The Misbehavior of Markets by Benoit Mandelbrot


Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, 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 von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, new economy, paper trading, passive investing, Paul Lévy, 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, volatility smile

His rival for the chair, Georges Cerf, was a brilliant young mathematician with all the right connections in Paris and an ally in Dijon, Maurice Gevrey, a sitting math professor. Gevrey appears to have taken a passionate dislike to Bachelier. Scouring the latter’s work, Gevrey soon spotted a glaring mathematical error. When the academic committee met to decide the professorship, Gevrey brandished a letter from the eminent French probabilist, Paul Lévy in Paris, confirming the fault. Result: “Bachelier was blackballed,” as Lévy ruefully recalled years later, in correspondence with me. By then, Lévy regretted the incident. He had read only the passage highlighted by Gevrey rather than the entire treatise; and in the full context of Bachelier’s work the error appears benign. Lévy later apologized to Bachelier that “an impression, produced by a single initial error, should have kept me from going on with my reading of a work in which there were so many interesting ideas.”

And he let fly a fusillade of more than four hundred vitriolic words at Lévy. He called Lévy’s critique “violent and unjustified” and based on total ignorance of his work. The Parisian, who had just finished a book on probability, had not even bothered “opening my book” on the subject before writing his own, Bachelier complained. He concluded with an insinuation typical of the time: “Without doubt, it is inconceivable that M. Paul Lévy had wanted, by a sort of last-minute trick, to favor un coreligionnaire.” Lévy was a Jew. Given Bachelier’s temper, it is remarkable that he ever won the security of a professorial chair—which he ultimately did, at Besançon. But that was twenty-seven years after his doctoral thesis, the work for which he is so well remembered today. The Coin-Tossing View of Finance The Bourse, the bustling Paris exchange, was at that time a world capital of bond trading.

Denying them would create an additional and totally unnecessary risk. Same formula, different result because of one change in the parameters. It is all exquisitely versatile. Clue No. 3: The Laws of Exceptional Chance The last hint in the cotton mystery goes back again to my student days. After the war, I was at the École Polytechnique, one of France’s “Ivy League,” the Grandes Écoles. One of my professors was Paul Lévy, a well-known mathematician, and the same man who had unintentionally played so decisive a role in Bachelier’s life story. Lévy was independently wealthy, the scion of a Jewish merchant and academic family. To students at the back of his lecture hall—as I was—he was near-inaudible and his long, gray, and well-groomed figure bore an odd resemblance to the somewhat peculiar way he had of tracing the long “?”

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, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, butterfly effect, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, 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, new economy, Paul Lévy, 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, V2 rocket, volatility smile

An event that occurred toward the end of Bachelier’s career, in 1926 (the year before he finally earned his permanent position), cast a pall over his final years as a teacher and may explain why he stopped publishing. That year, Bachelier applied for a permanent position at Dijon, where he had been teaching for several years. One of his colleagues, in reviewing his work, became confused by Bachelier’s notation. Believing he had found an error, he sent the document to Paul Lévy, a younger but more famous French probability theorist. Lévy, examining only the page on which the error purportedly appeared, confirmed the Dijon mathematician’s suspicions. Bachelier was blacklisted from Dijon. Later, he learned of Lévy’s part in the fiasco and became enraged. He circulated a letter claiming that Lévy had intentionally blocked his career without understanding his work. Bachelier earned his position at Besançon a year later, but the damage had been done and questions concerning the legitimacy of much of Bachelier’s work remained.

He quickly confirmed Houthakker’s most troubling findings: it appeared that there was no “average” rate of return. The prices looked random, but they weren’t explained by the standard statistical tools or Bachelier’s and Osborne’s theories. Something weird was going on. Mandelbrot had seen unusual distributions before. In addition to studying Zipf’s and Pareto’s work, he was familiar with a third kind of distribution, discovered by one of his professors in Paris, Paul Lévy. It was Lévy who, upon reading a small section of one of Bachelier’s papers, concluded that Bachelier’s work was plagued with errors. Much later, Lévy would recognize his own mistake and apologize to Bachelier. Part of what made Lévy return to Bachelier’s work was a renewed interest in random walk processes and probability distributions. Ironically, this later work of Lévy’s received far less attention than his earlier work, leaving Lévy alienated and obscure at the twilight of his career.

As described below, Mandelbrot would later argue that the distributions of rates of return for financial markets do have finite means, but not variances. However, it can often be difficult to calculate the mean for a Lévy-stable distribution — in cases where variance is undefined, the average value calculated from any finite data set takes a long time to converge to the mean — which accounts for why Mandelbrot and Houthakker originally believed that the mean did not exist. “. . . discovered by one of his professors in Paris, Paul Lévy”: Mandelbrot offers some biographical background on Lévy in Mandelbrot (1982) and describes his interactions with him in Mandelbrot and Hudson (2004). “. . . a class of probability distribution now called Lévy-stable distributions”: They are also called α-stable distributions. Throughout the text (and in Mandelbrot’s popular writing), “wildness” is code for “α < 2.” For a Lévy-stable distribution with 1 < α < 2, the mean is defined, but the variance is not; if α ≤ 1, neither mean nor variance is defined.

pages: 695 words: 194,693

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, 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,, 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: store of value / unit of account / medium of exchange, moral hazard, 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, transatlantic slave trade, tulip mania, wage slave

The “high priest” of non-normality before Nassim Taleb ever started to trade or write about extreme events was Benoit Mandelbrot, the creator of fractal geometry, a mathematician who both carried the mantle of French mathematical finance and who also believed he had discovered its fatal flaw. Mandelbrot was a student of Paul Lévy’s—the son of the man who gave Bachelier bad marks at his examination at the École Polytechnique in 1900. Lévy’s research focused on “stochastic processes”: mathematical models that describe the behavior of some variable through 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.

pages: 374 words: 114,600

The Quants by Scott Patterson


Albert Einstein, asset allocation, automated trading system, Benoit Mandelbrot, Bernie Madoff, Bernie Sanders, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, buttonwood tree, buy low sell high, capital asset pricing model, centralized clearinghouse, Claude Shannon: information theory, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Donald Trump, Doomsday Clock, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Gordon Gekko, greed is good, Haight Ashbury, index fund, invention of the telegraph, invisible hand, Isaac Newton, job automation, John Nash: game theory, law of one price, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, merger arbitrage, NetJets, new economy, offshore financial centre, Paul Lévy, Ponzi scheme, quantitative hedge fund, quantitative trading / quantitative finance, race to the bottom, random walk, Renaissance Technologies, risk-adjusted returns, 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

While praising Louis Bachelier, a personal hero of Mandelbrot’s, the mathematician asserted that “the empirical distributions of price changes are usually too ‘peaked’ relative to samples” from standard distributions. The reason: “Large price changes are much more frequent than predicted.” Mandelbrot proposed an alternative method to measure the erratic behavior of prices, one that borrows a mathematical technique devised by the French mathematician Paul Lévy, whom he’d studied under in Paris. Lévy investigated distributions in which a single sample radically changes the curve. The average of the heights of 1,000 people won’t change very much as a result of the height of the 1,001st person. But a so-called Lévy distribution can be thrown off by a single wild shift in the sample. Mandelbrot uses the example of a blindfolded archer: 1,000 shots may fall close to the target, but the 1,001st shot, by happenstance, may fall very wide of the mark, radically changing the overall distribution.

pages: 461 words: 128,421

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


Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, bank run, Benoit Mandelbrot, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, card file, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, 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, endowment effect, Eugene Fama: efficient market hypothesis, experimental economics, financial innovation, Financial Instability Hypothesis, floating exchange rates, George Akerlof, Henri Poincaré, Hyman Minsky, implied volatility, impulse control, index arbitrage, index card, index fund, invisible hand, Isaac Newton, John Nash: game theory, John von Neumann, joint-stock company, Joseph Schumpeter, libertarian paternalism, linear programming, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market design, New Journalism, Nikolai Kondratiev, Paul Lévy, 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 Shiller, Robert Shiller, rolodex, Ronald Reagan, shareholder value, Sharpe ratio, short selling, side project, Silicon Valley, South Sea Bubble, statistical model, 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, volatility smile, Yogi Berra

Bachelier went on to a modestly successful career as a math professor, and published a well-received popular treatise on games, chance, and risk (Le jeu, la chance et le hasard). When he died in 1946, one year before Irving Fisher, no one on the trading floor was making use of his ideas. His colleagues, meanwhile, were nonplussed by his interest in markets. On a bibliography of Bachelier’s writings found in the files of the great French mathematician Paul Lévy is scrawled the complaint, “Too much on finance!”8 IRVING FISHER WAS ABLE TO go where Bachelier did not because he had more than just mathematics and probability theory at his disposal. He was an economist. He was able to go where other economists did not because he, unlike all but a handful of them at the time, was a mathematician. And he was able to do something tangible with his insights because he was a wealthy resident of a country where, in the early decades of the twentieth century, financial markets were just beginning to grow into the vast bazaars that would steer the economy for the rest of the century and beyond.