Volatility Surface

JIM GATHERAL is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.Dr. Gatheral obtained a PhD in theoretical physics from Cambridge Universityin 1983. Since then, he has been involved in all of the major derivative product areasas a bookrunner, risk manager, and quantitative analyst in London, Tokyo, and New York. From 1997 to 2005, Dr. Gatheral headed the Equity Quantitative Analytics group at Merrill Lynch. His current research focus is equity market microstructure and algorithmic trading.With a foreword by Nassim Nicholas TalebTaleb is the Dean's Professor in the Sciences of Uncertainty at the University of Massachusetts at Amherst. He is also author of Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets (Random House, 2005).

“…I do recommend this book…” (Zentralblatt MATH , Vol. 1118 2007/20)

• List of Figures xiiiList of Tables xixForeword xxiPreface xxiiiAcknowledgments xxviiCHAPTER 1 Stochastic Volatility and Local Volatility 1Stochastic Volatility 1Derivation of the Valuation Equation 4Local Volatility 7History 7A Brief Review of Dupire’s Work 8Derivation of the Dupire Equation 9Local Volatility in Terms of Implied Volatility 11Special Case: No Skew 13Local Variance as a Conditional Expectation of Instantaneous Variance 13CHAPTER 2 The Heston Model 15The Process 15The Heston Solution for European Options 16A Digression: The Complex Logarithm in the Integration (2.13) 19Derivation of the Heston Characteristic Function 20Simulation of the Heston Process 21Milstein Discretization 22Sampling from the Exact Transition Law 23Why the Heston Model Is so Popular 24CHAPTER 3 The Implied Volatility Surface 25Getting Implied Volatility from Local Volatilities 25Model Calibration 25Understanding Implied Volatility 26Local Volatility in the Heston Model 31Ansatz 32Implied Volatility in the Heston Model 33The Term Structure of Black-Scholes Implied Volatility in the Heston Model 34The Black-Scholes Implied Volatility Skew in the Heston Model 35The SPX Implied Volatility Surface 36Another Digression: The SVI Parameterization 37A Heston Fit to the Data 40Final Remarks on SV Models and Fitting the Volatility Surface 42CHAPTER 4 The Heston-Nandi Model 43Local Variance in the Heston-Nandi Model 43A Numerical Example 44The Heston-Nandi Density 45Computation of Local Volatilities 45Computation of Implied Volatilities 46Discussion of Results 49CHAPTER 5 Adding Jumps 50Why Jumps are Needed 50Jump Diffusion 52Derivation of the Valuation Equation 52Uncertain Jump Size 54Characteristic Function Methods 56Lévy Processes 56Examples of Characteristic Functions for Specific Processes 57Computing Option Prices from the Characteristic Function 58Proof of (5.6) 58Computing Implied Volatility 60Computing the At-the-Money Volatility Skew 60How Jumps Impact the Volatility Skew 61Stochastic Volatility Plus Jumps 65Stochastic Volatility Plus Jumps in the Underlying Only (SVJ) 65Some Empirical Fits to the SPX Volatility Surface 66Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ) 68SVJ Fit to the September 15, 2005, SPX Option Data 71Why the SVJ Model Wins 73CHAPTER 6 Modeling Default Risk 74Merton’s Model of Default 74Intuition 75Implications for the Volatility Skew 76Capital Structure Arbitrage 77Put-Call Parity 77The Arbitrage 78Local and Implied Volatility in the Jump-to-Ruin Model 79The Effect of Default Risk on Option Prices 82The CreditGrades Model 84Model Setup 84Survival Probability 85Equity Volatility 86Model Calibration 86CHAPTER 7 Volatility Surface Asymptotics 87Short Expirations 87The Medvedev-Scaillet Result 89The SABR Model 91Including Jumps 93Corollaries 94Long Expirations: Fouque, Papanicolaou, and Sircar 95Small Volatility of Volatility: Lewis 96Extreme Strikes: Roger Lee 97Example: Black-Scholes 99Stochastic Volatility Models 99Asymptotics in Summary 100CHAPTER 8 Dynamics of the Volatility Surface 101Dynamics of the Volatility Skew under Stochastic Volatility 101Dynamics of the Volatility Skew under Local Volatility 102Stochastic Implied Volatility Models 103Digital Options and Digital Cliquets 103Valuing Digital Options 104Digital Cliquets 104CHAPTER 9 Barrier Options 107Definitions 107Limiting Cases 108Limit Orders 108European Capped Calls 109The Reflection Principle 109The Lookback Hedging Argument 112One-Touch Options Again 113Put-Call Symmetry 113QuasiStatic Hedging and Qualitative Valuation 114Out-of-the-Money Barrier Options 114One-Touch Options 115Live-Out Options 116Lookback Options 117Adjusting for Discrete Monitoring 117Discretely Monitored Lookback Options 119Parisian Options 120Some Applications of Barrier Options 120Ladders 120Ranges 120Conclusion 121CHAPTER 10 Exotic Cliquets 122Locally Capped Globally Floored Cliquet 122Valuation under Heston and Local Volatility Assumptions 123Performance 124Reverse Cliquet 125Valuation under Heston and Local Volatility Assumptions 126Performance 127Napoleon 127Valuation under Heston and Local Volatility Assumptions 128Performance 130Investor Motivation 130More on Napoleons 131CHAPTER 11 Volatility Derivatives 133Spanning Generalized European Payoffs 133Example: European Options 134Example: Amortizing Options 135The Log Contract 135Variance and Volatility Swaps 136Variance Swaps 137Variance Swaps in the Heston Model 138Dependence on Skew and Curvature 138The Effect of Jumps 140Volatility Swaps 143Convexity Adjustment in the Heston Model 144Valuing Volatility Derivatives 146Fair Value of the Power Payoff 146The Laplace Transform of Quadratic Variation under Zero Correlation 147The Fair Value of Volatility under Zero Correlation 149A Simple Lognormal Model 151Options on Volatility: More on Model Independence 154Listed Quadratic-Variation Based Securities 156The VIX Index 156VXB Futures 158Knock-on Benefits 160Summary 161Postscript 162Bibliography 163Index 169