INSEAD Library

Includes bibliographical references and index

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The Volatility Surface A Practioner's Guide Contents List of Figures List of Tables Foreword Preface Acknowledgments CHAPTER 1 Stochastic Volatility and Local Volatility Stochastic Volatility Derivation of the Valuation Equation Local Volatility History A Brief Review of Dupire's Work Derivation of the Dupire Equation Local Volatility in Terms of Implied Volatility Special Case: No Skew Local Variance as a Conditional Expectation of Instantaneous Variance xiii xix xxi xxiii xxvii 1 1 4 7 7 8 9 11 13 13 CHAPTER 2 The Heston Model The Process The Heston Solution for European Options A Digression: The Complex Logarithm in the Integration (2.13) Derivation of the Heston Characteristic Function Simulation of the Heston Process Milstein Discretization Sampling from the Exact Transition Law Why the Heston Model Is so Popular 15 15 16 19 20 21 22 23 24 CHAPTER 3 The implied Volatility Surface Getting Implied Volatility from Local Volatilities Model Calibration Understanding Implied Volatility Local Volatility in the Heston Model Ansatz Implied Volatility in the Heston Model The Term Structure of Black-Scholes Implied Volatility in the Heston Model The Black-Scholes Implied Volatility Skew in the Heston Model The SPX Implied Volatility Surface Another Digression: The SVI Parameterization A Heston Fit to the Data Final Remarks on SV Models and Fitting the Volatility Surface 25 25 25 26 31 32 33 34 35 36 37 40 42 CHAPTER 4 The Heston-Nandi Model Local Variance in the Heston-Nandi Model A Numerical Example The Heston-Nandi Density Computation of Local Volatilities Computation of Implied Volatilities Discussion of Results 48 43 44 45 45 46 49 CHAPTER 5 Adding Jumps Why Jumps are Needed Jump Diffusion Derivation of the Valuation Equation Uncertain Jump Size Characteristic Function Methods Levy Processes Examples of Characteristic Functions for Specific Processes Computing Option Prices from the Characteristic Function Proof of (5.6) 50 50 52 52 54 56 56 57 58 58 Computing Implied Volatility Computing the At-the-Money Volatility Skew How Jumps Impact the Volatility Skew Stochastic Volatility Plus Jumps Stochastic Volatility Plus Jumps in the Underlying Only (SVJ) Some Empirical Fits to the SPX Volatility Surface Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ) SVJ Fit to the September 15, 2005, SPX Option Data Why the SVJ Model Wins 60 60 61 65 65 66 68 71 73 CHAPTER 8 Modeling Default Risk Merton's Model of Default Intuition Implications for the Volatility Skew Capital Structure Arbitrage Put-Call Parity The Arbitrage Local and Implied Volatility in the Jump-to-Ruin Model The Effect of Default Risk on Option Prices The CreditGrades Model Model Setup Survival Probability Equity Volatility Model Calibration 74 74 75 76 77 77 78 79 82 84 84 85 86 86 CHAPTER 7 Volatility Surface Asymptotics Short Expirations The Medvedev-Scaillet Result The SABR Model Including Jumps Corollaries Long Expirations: Fouque, Papanicolaou, and Sircar Small Volatility of Volatility: Lewis Extreme Strikes: Roger Lee Example: Black-Scholes Stochastic Volatility Models Asymptotics in Summary 87 87 89 91 93 94 95 96 97 99 99 100 CHAPTER 8 Dynamics of the Volatility Surface Dynamics of the Volatility Skew under Stochastic Volatility Dynamics of the Volatility Skew under Local Volatility Stochastic Implied Volatility Models Digital Options and Digital Cliquets Valuing Digital Options Digital Cliquets 101 101 102 103 103 104 104 CHAPTER 8 Barrier Options Definitions Limiting Cases Limit Orders European Capped Calls The Reflection Principle The Lookback Hedging Argument One-Touch Options Again Put-Call Symmetry QuasiStatic Hedging and Qualitative Valuation Out-of-the-Money Barrier Options One-Touch Options Live-Out Options Lookback Options Adjusting for Discrete Monitoring Discretely Monitored Lookback Options Parisian Options Some Applications of Barrier Options Ladders Ranges Conclusion 107 107 108 108 109 109 112 113 113 114 114 115 116 117 117 119 120 120 120 120 121 CHAPTER 10 Exotic Cliquets Locally Capped Globally Floored Cliquet Valuation under Heston and Local Volatility Assumptions Performance Reverse Cliquet 122 122 123 124 125 Valuation under Heston and Local Volatility Assumptions Performance Napoleon Valuation under Heston and Local Volatility Assumptions Performance Investor Motivation More on Napoleons 126 127 127 128 130 130 131 CHAPTER 11 Volatility Derivatives Spanning Generalized European Payoffs Example: European Options Example: Amortizing Options The Log Contract Variance and Volatility Swaps Variance Swaps Variance Swaps in the Heston Model Dependence on Skew and Curvature The Effect of Jumps Volatility Swaps Convexity Adjustment in the Heston Model Valuing Volatility Derivatives Fair Value of the Power Payoff The Laplace Transform of Quadratic Variation under Zero Correlation The Fair Value of Volatility under Zero Correlation A Simple Lognormal Model Options on Volatility: More on Model Independence Listed Quadratic-Variation Based Securities The VIX Index VXB Futures Knock-on Benefits Summary 188 133 134 135 135 136 137 138 138 140 143 144 146 146 147 149 151 154 156 156 158 160 161 Postscript Bibliography Index 162 163 180