272 pages
October 2005
Illustrations.
Acknowledgments.
Introduction.
Summary.
Contributions and Further Research.
Data and Programs.
Chapter 1: The Volatility Problem.
Introduction.
The Stock Market.
The Stock Price Process.
Historic Volatility.
The Derivatives Market.
The Black-Scholes Approach.
The Cox-Ross-Rubinstein Approach.
Jump Diffusion and Level-Dependent Volatility.
Jump Diffusion.
Level-Dependent Volatility.
Local Volatility.
The Dupire Approach.
The Derman-Kani Approach.
Stability Issues.
Calibration Frequency.
Stochastic Volatility.
Stochastic Volatility Processes.
GARCH and Diffusion Limits.
The Pricing PDE Under Stochastic Volatility.
The Market Price of Volatility Risk.
The Two-Factor PDE.
The Generalized Fourier Transform.
The Transform Technique.
Special Cases.
The Mixing Solution.
The Romano-Touzi Approach.
A One-Factor Monte Carlo Technique.
The Long-Term Asymptotic Case.
The Deterministic Case.
The Stochastic Case.
A Series Expansion on Volatility-of-Volatility.
Pure-Jump Models.
Variance Gamma.
Variance Gamma with Stochastic Arrival.
Variance Gamma with Gamma Arrival Rate.
Chapter 2: The Inference Problem.
Introduction.
Using Option Prices.
Direction Set (Powell) Method.
Numeric Tests.
The Distribution of the Errors.
Using Stock Prices.
The Likelihood Function.
Filtering.
The Simple and Extended Kalman Filters.
The Unscented Kalman Filter.
Kushner’s Nonlinear Filter.
Parameter Learning.
Parameter Estimation via MLE.
Diagnostics.
Particle Filtering.
Comparing Heston with Other Models.
The Performance of the Inference Tools.
The Bayesian Approach.
Using the Characteristic Function.
Introducing Jumps.
Pure Jump Models.
Recapitulation.
Model Identification.
Convergence Issues and Solutions.
Chapter 3: The Consistency Problem.
Introduction.
The Consistency Test.
The Setting.
The Cross-Sectional Results.
Robustness Issues for the Cross-Sectional Method.
Time-Series Results.
Financial Interpretation.
The Peso Theory.
Background.
Numeric Results.
Trading Strategies.
Skewness Trades.
Kurtosis Trades.
Directional Risks.
An Exact Replication.
The Mirror Trades.
An Example of the Skewness Trade.
Multiple Trades.
High Volatility-of-Volatility and High Correlation.
Non-Gaussian Case.
VGSA.
AWord of Caution.
Foreign Exchange, Fixed Income, and Other Markets.
Foreign Exchange.
Fixed Income.
References.
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