CONTENTS. Introduction. Acknowledgments. What Is New in the Second Edition? Option Pricing Formulas Overview. Glossary of Notations

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1 Introduction Acknowledgments What Is New in the Second Edition? Option Pricing Formulas Overview Glossary of Notations xvii xix xxi xxiii xxxv 1 Black-Scholes-Merton Black-Scholes-Merton The Black-Scholes Option Pricing Formula Options on Stock Indexes Options on Futures Margined Options on Futures Currency Options The Generalized Black-Scholes-Merton Option Pricing Formula Parities and Symmetries Put-Call Parity for European Options At-the-Money Forward Value Symmetry Put-Call Symmetry Put-Call Supersymmetry Black-Scholes-Merton on Variance Form Before Black-Scholes-Merton The Bachelier Model The Sprenkle Model The Boness Model The Samuelson Model Appendix A: The Black-Scholes-Merton PDE 15 Haug, Espen Gaarder The complete guide to option pricing formulas 2007 vii digitalisiert durch: IDS Basel Bern

2 Vlll CONTENTS Ito's Lemma Dynamic Hedging 16 Black-Scholes-Merton Greeks Delta Greeks Delta Delta Mirror Strikes and Assets Strike from Delta Futures Delta from Spot Delta DdeltaDvol and DvegaDspot DvannaDvol DdeltaDtime, Charm Elasticity Gamma Greeks Gamma Maximal Gamma and the Illusions of Risk GammaP Gamma Symmetry DgammaDvol, Zomma DgammaDspot, Speed DgammaDtime, Color Vega Greeks Vega Vega Symmetry Vega-Gamma Relationship Vega from Delta VegaP Vega Leverage, Vega Elasticity DvegaDvol, Vomma DvommaDvol, Ultima DvegaDtime Variance Greeks Variance Vega DdeltaDvar Variance Vomma Variance Ultima Volatility-Time Greeks Theta Greeks Theta Theta Symmetry Rho Greeks Rho Phi/Rho Carry Rho 73

3 ix 2.8 Probability Greeks In-the-Money Probability DzetaDvol DzetaDtime Risk-Neutral Probability Density From in-the-money Probability to Density Probability ofever Getting in-the-money Greeks Aggregations Net Weighted Vega Exposure At-the-Money Forward Approximations Approximation of the Black-Scholes-Merton Formula Delta Gamma Vega Theta Rho Cost-of-Carry Numerical Greeks First-Order Greeks Second-Order Greeks Third-Order Greeks Mixed Greeks Third-Order Mixed Greeks Greeks from Closed-Form Approximations Appendix B: Taking Partial Derivatives 90 Analytical Formulas for American Options The Barone-Adesi and Whaley Approximation The Bjerksund and Stensland (1993) Approximation The Bjerksund and Stensland (2002) Approximation Put-Call Transformation American Options American Perpetual Options 109 Exotic Options Single Asset Variable Purchase Options Executive Stock Options Moneyness Options Power Contracts and Power Options Power Contracts Standard Power Option Capped Power Option Powered Option Log Contracts LogOS) Contract 120

4 4.5.2 Log Option Forward Start Options Fade-in Option Ratchet Options Reset Strike Options Type Reset Strike Options Type Time-Switch Options Chooser Options Simple Chooser Options Complex Chooser Options Options on Options Put-Call Parity Compound Options Compound Option Approximation Options with Extendible Maturities Options That Can Be Extended by the Holder Writer-Extendible Options Lookback Options Floating-Strike Lookback Options Fixed-Strike Lookback Options Partial-Time Floating-Strike Lookback Options Partial-Time Fixed-Strike Lookback Options Extreme-Spread Options Mirror Options Barrier Options Standard Barrier Options Standard American Barrier Options Double-Barrier Options Partial-Time Single-Asset Barrier Options Look-Barrier Options Discrete-Barrier Options Soft-Barrier Options Use of Put-Call Symmetry for Barrier Options Barrier Option Symmetries First-Then-Barrier Options Double-Barrier Option Using Barrier Symmetry Dual Double-Barrier Options Binary Options Gap Options Cash-or-Nothing Options Asset-or-Nothing Options Supershare Options Binary Barrier Options Double-Barrier Binary Options 180

5 xi Double-Barrier Binary Asymmetrical Asian Options Geometrie Average-Rate Options Arithmetic Average-Rate Options Discrete Arithmetic Average-Rate Options Equivalence of Floating-Strike and Fixed-Strike Asian Options Asian Options with Volatility Term-Structure Exotic Options on Two Assets Relative Outperformance Options Product Options Two-Asset Correlation Options Exchange-One-Asset-for-Another Options American Exchange-One-Asset-for-Another Option Exchange Options on Exchange Options Options on the Maximum or the Minimum of Two Risky Assets Spread-Option Approximation Two-Asset Barrier Options Partial-Time Two-Asset Barrier Options Margrabe Barrier Options Discrete-Barrier Options Two-Asset Cash-or-Nothing Options Best or Worst Cash-or-Nothing Options Options on the Minimum or Maximum of Two Averages Currency-Translated Options Foreign Equity Options Struck in Domestic Currency Fixed Exchange Rate Foreign Equity Options Equity Linked Foreign Exchange Options Takeover Foreign Exchange Options Greeks for Two-Asset Options Black-Scholes-Merton Adjustments and Alternatives The Black-Scholes-Merton Model with Delayed Settlement The Black-Scholes-Merton Model Adjusted for Trading Day Volatility Discrete Hedging Hedging Error Discrete-Time Option Valuation and Delta Hedging Discrete-Time Hedging with Transaction Cost.. 238

6 Xll CONTENTS 6.4 Option Pricing in Trending Markets Alternative Stochastic Processes Constant Elasticity of Variance Skewness-Kurtosis Models Definition of Skewness and Kurtosis The Skewness and Kurtosis for a Lognormal Distribution Jarrow and Rudd Skewness and Kurtosis Model The Corrado and Su Skewness and Kurtosis Model Modified Corrado-Su Skewness-Kurtosis Model Skewness-Kurtosis Put-Call Supersymmetry Skewness-Kurtosis Equivalent Black-Scholes-Merton Volatility Gram Charlier Density Skewness-Kurtosis Trees Pascal Distribution and Option Pricing Jump-Diffusion Models The Merton Jump-Diffusion Model Bates Generalized Jump-Diffusion Model Stochastic Volatility Models Hull-White Uncorrelated Stochastic Volatility Model Hull-White Correlated Stochastic Volatility Model The SABR Model Variance and Volatility Swaps Variance Swaps Volatility Swaps More Information 278 Trees and Finite Difference Methods Binomial Option Pricing Cox-Ross-Rubinstein American Binomial Tree Greeks in CRR Binomial Tree Rendleman Bartter Binomial Tree Leisen-Reimer Binomial Tree Convertible Bonds in Binomial Trees Binomial Model with Skewness and Kurtosis Trinomial Trees Exotic Options in Tree Models Options on Options Barrier Options Using Brownian Bridge Probabilities 305

7 xiii American Barrier Options in CRR Binomial Tree European Reset Options Binomial American Asian Options in a Tree Three-Dimensional Binomial Trees Implied Tree Models Implied Binomial Trees Implied Trinomial Trees Finite Difference Methods Explicit Finite Difference Implicit Finite Difference Finite Difference in ln(s) The Crank-Nicolson Method Monte Carlo Simulation Standard Monte Carlo Simulation Greeks in Monte Carlo Monte Carlo for Callable Options Two Assets Three Assets N Assets, Cholesky Decomposition Monte Carlo of Mean Reversion Generating Pseudo-Random Numbers Variance Reduction Techniques Antithetic Variance Reduction IQ-MC/Importance Sampling IQ-MC Two Correlated Assets Quasi-Random Monte Carlo American Option Monte Carlo Options on Stocks That Pay Discrete Dividends European Options on Stock with Discrete Cash Dividend The Escrowed Dividend Model Simple Volatility Adjustment Haug-Haug Volatility Adjustment Bos-Gairat-Shepeleva Volatility Adjustment Bos-Vandermark Non-Recombining Tree Black's Method for Calls on Stocks with Known Dividends The Roll, Geske, and Whaley Model Benchmark Model for Discrete Cash Dividend A Single Dividend Multiple Dividends 382

8 xiv CONTENTS Applications Options on Stocks with Discrete Dividend Yield European with Discrete Dividend Yield Closed-Form American Call Recombining Tree Model Commodity and Energy Options Energy Swaps/Forwards Energy Options Options on Forwards, Black-76F Energy Swaptions Hybrid Payoff Energy Swaptions The Miltersen-Schwartz Model Mean Reversion Model 10.5 Seasonality Interest Rate Derivatives FRAs and Money Market Instruments FRAs From Cash Deposits The Relationship between FRAs and Currency Forwards Convexity Adjustment Money Market Futures Simple Bond Mathematics Dirty and Clean Bond Price Current Yield Modified Duration and BPV Bond Price and Yield Relationship Price and Yield Relationship for a Bond From Bond Price to Yield Pricing Interest Rate Options Using Black Options on Money Market Futures Price and Yield Volatility in Money Market Futures Caps and Floors Swaptions Swaption Volatilities from Caps or FRA Volatilities Swaptions with Stochastic Volatility Convexity Adjustments European Short-Term Bond Options From Price to Yield Volatility in Bonds The Schaefer and Schwartz Model One-Factor Term Structure Models The Rendleman and Bartter Model The Vasicek Model 430

9 xv The Ho and Lee Model The Hüll and White Model The Black-Derman-Toy Model Volatility and Correlation Historical Volatility Historical Volatility from Close Prices High-Low Volatility High-Low-Close Volatility Exponential Weighted Historical Volatility From Annual Volatility to Daily Volatility Confidence Intervals for the Volatility Estimate Volatility Cones Implied Volatility The Newton-Raphson Method The Bisection Method Implied Volatility Approximations Implied Forward Volatility From Implied Volatility Surface to Local Volatility Surface Confidence Interval for the Asset Price Basket Volatility Historical Correlation Distribution of Historical Correlation Coefficient Implied Correlations Implied Correlation from Currency Options Average Implied Index Correlation Various Formulas Probability of High or Low, the Arctangent Rule Siegel's Paradox and Volatility Ratio Effect Distributions The Cumulative Normal Distribution Function The Hart Algorithm Polynomial Approximations The Inverse Cumulative Normal Distribution Function The Bivariate Normal Density Function The Cumulative Bivariate Normal Distribution Function The Trivariate Cumulative Normal Distribution Function 482

10 XVI CONTENTS 14 Some Useful Formulas Interpolation Linear Interpolation Log-Linear Interpolation Exponential Interpolation Cubic Interpolation: Lagrange's Formula Cubic-Spline Interpolation Two-Dimensional Interpolation Interest Rates Future Value of Annuity Net Present Value of Annuity Continuous Compounding Compounding Frequency Zero-Coupon Rates from Par Bonds/Par Swaps Risk-Reward Measures Treynor's Measure Sharpe Ratio Confidence Ratio Sortino Ratio Burke Ratio Return on VaR Jensen's Measure Appendix C: Basic Useful Information 496 The Option Pricing Software 497 Bibliography 499 Index 521