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 |
List of Figures |
List of Tables |
Foreword |
Preface |
Acknowledgments |
Stochastic Volatility and Local Volatility / Chapter 1: |