The Volatility Surface: A Practitioner's Guide
by Jim Gatheral; Foreword by: Nassim Nicholas TalebFormat: eBook
Pub. Date: 2006-12-01
Publisher(s): Wiley
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Summary
--Bruno Dupire, winner of the 2006 Wilmott Cutting Edge Research Award Quantitative Research, Bloomberg LP
Table of Contents
| List of Figures | |
| List of Tables | |
| Foreword | |
| Preface | |
| Acknowledgments | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| Adding Jumps | |
| Why Jumps are Needed | |
| Jump Diffusion | |
| Derivation of the Valuation Equation | |
| 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 | |
| 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 | |
| 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 | |
| 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 Cliq | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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