| List of Figures |
| List of Tables |
| Foreword |
| Preface |
| Acknowledgments |
| Stochastic Volatility and Local Volatility / Chapter 1： |
| 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 |
| The Heston Model / Chapter 2： |
| 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 |
| The Implied Volatility Surface / Chapter 3： |
| 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 |
| The Heston-Nandi Model / Chapter 4： |
| 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 |
| Adding Jumps / Chapter 5： |
| Why Jumps are Needed |
| Jump Diffusion |
| Uncertain Jump Size |
| Characteristic Function Methods |
| L'evy Processes |
| Examples of Characteristic Functions for Specific Processes |
| Computing Option Prices from the Characteristic Function |
| Proof of (5.6) |
| 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 |
| Modeling Default Risk / Chapter 6： |
| 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 |
| Volatility Surface Asymptotics / Chapter 7： |
| 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 |
| Dynamics of the Volatility Surface / Chapter 8： |
| Dynamics of the Volatility Skew under Stochastic Volatility |
| Dynamics of the Volatility Skew under Local Volatility |
| Stochastic Implied Volatility Models |
| Digital Options and Digital Cliq |
