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List of Figures | |
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List of Tables | |
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Foreword | |
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Preface | |
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Acknowledgments | |
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Stochastic Volatility and Local Volatility | |
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Stochastic Volatility | |
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Derivation of the Valuation Equation | |
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Local Volatility | |
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History | |
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A Brief Review of Dupire's Work | |
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Derivation of the Dupire Equation | |
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Local Volatility in Terms of Implied Volatility | |
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Special Case: No Skew | |
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Local Variance as a Conditional Expectation of Instantaneous Variance | |
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The Heston Model | |
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The Process | |
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The Heston Solution for European Options | |
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A Digression: The Complex Logarithm in the Integration (2.13) | |
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Derivation of the Heston Characteristic Function | |
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Simulation of the Heston Process | |
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Milstein Discretization | |
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Sampling from the Exact Transition Law | |
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Why the Heston Model Is so Popular | |
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The Implied Volatility Surface | |
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Getting Implied Volatility from Local Volatilities | |
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Model Calibration | |
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Understanding Implied Volatility | |
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Local Volatility in the Heston Model | |
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Ansatz | |
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Implied Volatility in the Heston Model | |
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The Term Structure of Black-Scholes Implied Volatility in the Heston Model | |
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The Black-Scholes Implied Volatility Skew in the Heston Model | |
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The SPX Implied Volatility Surface | |
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Another Digression: The SVI Parameterization | |
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A Heston Fit to the Data | |
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Final Remarks on SV Models and Fitting the Volatility Surface | |
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The Heston-Nandi Model | |
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Local Variance in the Heston-Nandi Model | |
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A Numerical Example | |
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The Heston-Nandi Density | |
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Computation of Local Volatilities | |
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Computation of Implied Volatilities | |
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Discussion of Results | |
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Adding Jumps | |
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Why Jumps are Needed | |
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Jump Diffusion | |
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Derivation of the Valuation Equation | |
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Uncertain Jump Size | |
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Characteristic Function Methods | |
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L'evy Processes | |
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Examples of Characteristic Functions for Specific Processes | |
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Computing Option Prices from the Characteristic Function | |
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Proof of (5.6) | |
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Computing Implied Volatility | |
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Computing the At-the-Money Volatility Skew | |
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How Jumps Impact the Volatility Skew | |
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Stochastic Volatility Plus Jumps | |
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Stochastic Volatility Plus Jumps in the Underlying Only (SVJ) | |
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Some Empirical Fits to the SPX Volatility Surface | |
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Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ) | |
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SVJ Fit to the September 15, 2005, SPX Option Data | |
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Why the SVJ Model Wins | |
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Modeling Default Risk | |
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Merton's Model of Default | |
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Intuition | |
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Implications for the Volatility Skew | |
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Capital Structure Arbitrage | |
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Put-Call Parity | |
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The Arbitrage | |
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Local and Implied Volatility in the Jump-to-Ruin Model | |
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The Effect of Default Risk on Option Prices | |
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The CreditGrades Model | |
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Model Setup | |
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Survival Probability | |
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Equity Volatility | |
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Model Calibration | |
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Volatility Surface Asymptotics | |
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Short Expirations | |
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The Medvedev-Scaillet Result | |
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The SABR Model | |
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Including Jumps | |
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Corollaries | |
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Long Expirations: Fouque, Papanicolaou, and Sircar | |
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Small Volatility of Volatility: Lewis | |
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Extreme Strikes: Roger Lee | |
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Example: Black-Scholes | |
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Stochastic Volatility Models | |
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Asymptotics in Summary | |
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Dynamics of the Volatility Surface | |
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Dynamics of the Volatility Skew under Stochastic Volatility | |
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Dynamics of the Volatility Skew under Local Volatility | |
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Stochastic Implied Volatility Models | |
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Digital Options and Digital Cliq | |