volatility arbitrage

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pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Abraham Wald, 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, currency risk, delta neutral, discrete time, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial engineering, 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, power law, quantitative trading / quantitative finance, random walk, regulatory arbitrage, risk free rate, 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

Mahalo and aloha to my ever-encouraging wife, Andrea. About the author Paul Wilmott is one of the most well-known names in derivatives and risk management. His academic and practitioner credentials are impeccable, having written over 100 research papers on mathematics and finance, and having been a partner in a highly profitable volatility arbitrage hedge fund. Dr Wilmott is a consultant, publisher, author and trainer, the proprietor of wilmott.com and the founder of the Certificate in Quantitative Finance (7city.com/cqf). He is the Editor in Chief of the bimonthly quant magazine Wilmott and the author of the student text Paul Wilmott Introduces Quantitative Finance, which covers classical quant finance from the ground up, and Paul Wilmott on Quantitative Finance, the three-volume research-level epic.

Risk magazine 7 (2) 32-39 (February) Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January) Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343 Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Lewis, A 2000 Option valuation under Stochastic Volatility. Finance Press Lyons, TJ 1995 Uncertain Volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance 2 117-133 Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition.

Jump-diffusion models allow the stock (and even the volatility) to be discontinuous. Such models contain so many parameters that calibration can be instantaneously more accurate (if not necessarily stable through time). References and Further Reading Gatheral, J 2006 The Volatility Surface. John Wiley & Sons Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Taylor, SJ & Xu, X 1994 The magnitude of implied volatility smiles: theory and empirical evidence for exchange rates. The Review of Futures Markets 13 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition. John Wiley & Sons What is GARCH? Short Answer GARCH stands for Generalized Auto Regressive Conditional Heteroscedasticity.


The Volatility Smile by Emanuel Derman,Michael B.Miller

Albert Einstein, Asian financial crisis, Benoit Mandelbrot, Black Monday: stock market crash in 1987, book value, Brownian motion, capital asset pricing model, collateralized debt obligation, continuous integration, Credit Default Swap, credit default swaps / collateralized debt obligations, discrete time, diversified portfolio, dividend-yielding stocks, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial engineering, fixed income, implied volatility, incomplete markets, law of one price, London Whale, mandelbrot fractal, market bubble, market friction, Myron Scholes, prediction markets, quantitative trading / quantitative finance, risk tolerance, riskless arbitrage, Sharpe ratio, statistical arbitrage, stochastic process, stochastic volatility, transaction costs, volatility arbitrage, volatility smile, Wiener process, yield curve, zero-coupon bond

In the limit, as the number of periods n → ∞ with n𝛿t = tn − t0 ≡ T remaining fixed, we can replace the sums by integrals to obtain the result CT − ΔT ST + Δ0 S0 erT + T ∫0 er(T−x) Sx [dΔx ]b (5.15) Here we have replaced the subscript n with T, to clearly indicate that these are the values at expiration. The subscript b at the end of the formula denotes a backward Itô integral2 in which the increment dΔx is the infinitesimal 1 The following sections are based on Riaz Ahmad and Paul Wilmott, “Which Free Lunch Would You Like Today Sir?: Delta Hedging, Volatility, Arbitrage and Optimal Portfolios” (2005). This chapter also owes a debt to Peter Carr, “Frequently Asked Questions in Option Pricing Theory” (1999). 2 For a review of backward Itô integrals, see Appendix B. 91 −Δ2 Short Δ2 – Δ1 shares. ... . Short Δn – Δn–1 shares. 2 ... . n −Δn ... . −Δ1 2 −Δ1 Short Δ1 − Δ0 shares to rehedge. 1 −Δ0 Net # of Shares −Δ0 Short Δ0 shares.

This single, theoretically unique implied tree will value all standard options in agreement with their market prices, and consistently within a single model, rather than having to use an inconsistent BSM framework with different underlying volatilities for each standard option. The local volatility surface calculated from market prices can also be useful for volatility arbitrage trading. You can calculate future local volatilities implied from option prices and then decide if they seem reasonable. If these future volatilities seem unreasonably low or high, you might consider buying or selling butterfly and calendar spreads, in effect betting on future realized volatility at some future stock level and time.

To describe the volatility smile in the presence of negative jumps, we could use similar reasoning applied to a put that, after the jump, would have a value close to that of a short position in a forward contract. References Ahmad, Riaz, and Paul Wilmott. 2005. “Which Free Lunch Would You Like Today Sir?: Delta Hedging, Volatility, Arbitrage and Optimal Portfolios.” Wilmott (November). Andersen, Leif, and Jesper Andreasen. 2000. “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing.” Review of Derivatives Research 4 (3): 231–262. Birru, Justin, and Stephen Figlewski. 2012. “Anatomy of a Meltdown: The Risk Neutral Density for the S&P 500 in the Fall of 2008.”


Trading Risk: Enhanced Profitability Through Risk Control by Kenneth L. Grant

backtesting, business cycle, buy and hold, commodity trading advisor, correlation coefficient, correlation does not imply causation, delta neutral, diversification, diversified portfolio, financial engineering, fixed income, frictionless, frictionless market, George Santayana, global macro, implied volatility, interest rate swap, invisible hand, Isaac Newton, John Meriwether, Long Term Capital Management, managed futures, market design, Myron Scholes, performance metric, price mechanism, price stability, proprietary trading, risk free rate, risk tolerance, risk-adjusted returns, Sharpe ratio, short selling, South Sea Bubble, Stephen Hawking, the scientific method, The Wealth of Nations by Adam Smith, transaction costs, two-sided market, uptick rule, value at risk, volatility arbitrage, yield curve, zero-coupon bond

However, there are scenarios under which these portfolios (perhaps due to the demands of the firms that provide leverage to convertible 106 TRADING RISK arbitrage portfolio managers) might suffer from the worst of all possible combinations: The bonds drop in value while the equities experience a contemporaneous rise. It behooves those who either manage or fund convertible arbitrage portfolios to have a clear understanding of the risks associated with these worst-case scenarios. 3. Options Volatility Arbitrage. For those intrepid few wishing to capitalize on subtle and theoretically unsustainable discrepancies in the volatility pricing of options with the same or highly similar underliers, it is prudent to create a scenario analysis that specifically targets conditions under which these mispricings extend themselves.

See Scientific method Optimal f, 245–251 Optimism, importance of, 4 Options: asymmetric payoff functions, 150–151 implications of, generally, 148–149 implied volatility, 86–89, 150 leverage, 151–153 nonlinear pricing dynamics, 149 pricing, 88–89, 106 strike price/underlying price, relationship between, 149–150 volatility arbitrage, 106 Out-of-the-money option, 150 Over-the-counter derivatives, 148 Performance analysis, 7–8 Performance metrics, 16, 35 Performance objectives: “going to the beach,” 32–36 importance of, 19–20, 29 nominal target return, 20, 24–26 optimal target return, 20–24 stop-out level, 20–21, 26–32 Performance ratio, 188–200 Performance success metrics: accuracy ratio (win/loss), 184–186 impact ratio, 186–188 performance ratio, 188–200 profitability concentration (90/10) ratio, 200–208 Planning, importance of, 9.


pages: 1,088 words: 228,743

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

Alan Greenspan, Andrei Shleifer, asset allocation, asset-backed security, availability heuristic, backtesting, balance sheet recession, bank run, banking crisis, barriers to entry, behavioural economics, Bernie Madoff, Black Swan, Bob Litterman, bond market vigilante , book value, Bretton Woods, business cycle, buy and hold, buy low sell high, capital asset pricing model, capital controls, carbon credits, Carmen Reinhart, central bank independence, classic study, collateralized debt obligation, commoditize, commodity trading advisor, corporate governance, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, currency risk, deal flow, debt deflation, deglobalization, delta neutral, demand response, discounted cash flows, disintermediation, diversification, diversified portfolio, dividend-yielding stocks, equity premium, equity risk 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 macro, 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, inverted yield curve, invisible hand, John Bogle, junk bonds, Kenneth Rogoff, laissez-faire capitalism, law of one price, London Interbank Offered Rate, Long Term Capital Management, loss aversion, low interest rates, managed futures, 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, pension time bomb, performance metric, Phillips curve, Ponzi scheme, prediction markets, price anchoring, price stability, principal–agent problem, private sector deleveraging, proprietary trading, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, random walk, reserve currency, Richard Thaler, risk free rate, risk tolerance, risk-adjusted returns, risk/return, riskless arbitrage, Robert Shiller, savings glut, search costs, selection bias, seminal paper, Sharpe ratio, short selling, sovereign wealth fund, statistical arbitrage, statistical model, stochastic volatility, stock buybacks, stocks for the long run, survivorship bias, systematic trading, tail risk, 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

Excess return for such a product is proportional to the difference between squared implied volatility and squared realized volatility over the life of the contract. Backtested results were extremely impressive until 2007 but the losses in autumn 2008 were dramatic—and traumatic, prompting many investors to leave these strategies. The Merrill Lynch volatility arbitrage strategy, shown below, lost 12 years’ gradually earned excess returns in less than two months. (All index volatility-selling strategies plummeted in autumn 2008, but leverage made the losses of this index exceptionally high.) Although implied volatilities were high, realized volatilities exceeded them, reaching levels only seen during the 1987 crash.

Performance and risk statistics of S&P and index option-trading strategies, 1989–2009 Sources: Bloomberg, Chicago Board of Exchange, Bank of America Merrill Lynch. Table 15.1 (based on weekly data) illustrates how 2008 “revealed” the riskiness of various option-trading strategies, while at the same time long-run performance statistics became less appealing. The downfall was even worse for the “pure” volatility arbitrage strategy than for covered-option-writing strategies. Over 1989–2009, the latter still display higher returns and Sharpe ratios than a simple long-equities strategy (first column), but this advantage comes with less appealing tail risk or higher moment exposures: more negative skewness and higher kurtosis.

This composite trend style index applies a simple trend-following rule each week on commodity futures, equity country futures, bond/rate futures, and foreign exchange forwards: go long (short) if the current price is above (below) the 12-month moving average. Data are mainly from Bloomberg. Volatility. Volatility-selling returns are based on the simulated Merrill Lynch Equity Volatility Arbitrage Index since 1989, which tries to gain from the typically positive gap between market-implied volatility and subsequent realized volatility of the S&P 500 index’s Bloomberg ticker MLHFEV1 index. Chapter 15 also analyzes the simulated covered-call-writing indices and put-writing indices (BXM, BXY, PUT in Bloomberg) on the S&P 500 index from the Chicago Board of Exchange, dating back to the 1980s.


pages: 590 words: 153,208

Wealth and Poverty: A New Edition for the Twenty-First Century by George Gilder

accelerated depreciation, affirmative action, Albert Einstein, Bear Stearns, Bernie Madoff, book value, British Empire, business cycle, capital controls, clean tech, cloud computing, collateralized debt obligation, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, deindustrialization, diversified portfolio, Donald Trump, equal pay for equal work, floating exchange rates, full employment, gentrification, George Gilder, Gunnar Myrdal, Home mortgage interest deduction, Howard Zinn, income inequality, independent contractor, inverted yield curve, invisible hand, Jane Jacobs, Jeff Bezos, job automation, job-hopping, Joseph Schumpeter, junk bonds, knowledge economy, labor-force participation, longitudinal study, low interest rates, margin call, Mark Zuckerberg, means of production, medical malpractice, Michael Milken, minimum wage unemployment, Money creation, money market fund, money: store of value / unit of account / medium of exchange, Mont Pelerin Society, moral hazard, mortgage debt, non-fiction novel, North Sea oil, paradox of thrift, Paul Samuelson, plutocrats, Ponzi scheme, post-industrial society, power law, price stability, Ralph Nader, rent control, Robert Gordon, Robert Solow, Ronald Reagan, San Francisco homelessness, scientific management, Silicon Valley, Simon Kuznets, Skinner box, skunkworks, Solyndra, Steve Jobs, The Wealth of Nations by Adam Smith, Thomas L Friedman, upwardly mobile, urban renewal, volatility arbitrage, War on Poverty, women in the workforce, working poor, working-age population, yield curve, zero-sum game

Let’s say you floated the hour, 60 minutes in an hour one day, 50 the next, 85 the next. You would soon have to have hedges to insure against changes in the measure of time,” just to calculate your hours of work in “real terms.” You would have runaway sales of “hour insurance swaps,” and GDP might even go up for awhile, but real economic progress is not volatility arbitrage. Or, to change the metaphor, U.S. monetary policy resembles a housing policy pronouncement: “If we change the size of a foot from 12 to 15 inches, everyone will have a bigger house.” As Forbes comments, “In the real world, you’ll likely end up with a lot of confusion and fewer homes being built.


pages: 1,544 words: 391,691

Corporate Finance: Theory and Practice by Pierre Vernimmen, Pascal Quiry, Maurizio Dallocchio, Yann le Fur, Antonio Salvi

"Friedman doctrine" OR "shareholder theory", accelerated depreciation, accounting loophole / creative accounting, active measures, activist fund / activist shareholder / activist investor, AOL-Time Warner, ASML, asset light, bank run, barriers to entry, Basel III, Bear Stearns, Benoit Mandelbrot, bitcoin, Black Swan, Black-Scholes formula, blockchain, book value, business climate, business cycle, buy and hold, buy low sell high, capital asset pricing model, carried interest, collective bargaining, conceptual framework, corporate governance, correlation coefficient, credit crunch, Credit Default Swap, currency risk, delta neutral, dematerialisation, discounted cash flows, discrete time, disintermediation, diversification, diversified portfolio, Dutch auction, electricity market, equity premium, equity risk premium, Eugene Fama: efficient market hypothesis, eurozone crisis, financial engineering, financial innovation, fixed income, Flash crash, foreign exchange controls, German hyperinflation, Glass-Steagall Act, high net worth, impact investing, implied volatility, information asymmetry, intangible asset, interest rate swap, Internet of things, inventory management, invisible hand, joint-stock company, joint-stock limited liability company, junk bonds, Kickstarter, lateral thinking, London Interbank Offered Rate, low interest rates, mandelbrot fractal, margin call, means of production, money market fund, moral hazard, Myron Scholes, new economy, New Journalism, Northern Rock, performance metric, Potemkin village, quantitative trading / quantitative finance, random walk, Right to Buy, risk free rate, risk/return, shareholder value, short selling, Social Responsibility of Business Is to Increase Its Profits, sovereign wealth fund, Steve Jobs, stocks for the long run, supply-chain management, survivorship bias, The Myth of the Rational Market, time value of money, too big to fail, transaction costs, value at risk, vertical integration, volatility arbitrage, volatility smile, yield curve, zero-coupon bond, zero-sum game

The illustrative financial analysis of the Italian appliance manufacturer Indesit will guide you throughout this section of the book. Section II reviews the basic theoretical knowledge you will need to make an assessment of the value of the firm. Here again, the emphasis is on reasoning, which in many cases will become automatic (Chapters 15–19): efficient capital markets, the time value of money, the price of risk, volatility, arbitrage, return, portfolio theory, present value and future value, market risk, beta, etc. Then we review the major types of financial securities: equity, debt and options, for the purposes of valuation, along with the techniques for issuing and placing them (Chapters 20–25). Section III is devoted to value, to its theoretical foundations and to its computation.