CHAPTER 1
The Volatility Problem 1
Introduction 1
The Stock Market 2
The Stock Price Process 2
Historic Volatility 3
The Derivatives Market 4
The Black-Scholes Approach 5
The Cox-Ross-Rubinstein Approach 6
Jump Diffusion and Level-Dependent Volatility 7
Jump Diffusion 8
Level-Dependent Volatility 10
Local Volatility 14
The Dupire Approach 14
The Derman-Kani Approach 17
Stability Issues 18
Calibration Frequency 19
Stochastic Volatility 20
Stochastic Volatility Processes 20
GARCH and Diffusion Limits 21
The Pricing PDE Under Stochastic Volatility 24
The Market Price of Volatility Risk 25
The Two-Factor PDE 26
The Generalized Fourier Transform 27
The Transform Technique 27
Special Cases 28
The Mixing Solution 30
The Romano-Touzi Approach 30

A One-Factor Monte Carlo Technique 32
The Long-Term Asymptotic Case 34
The Deterministic Case 34
The Stochastic Case 35
A Series Expansion on Volatility-of-Volatility 37
Pure-Jump Models 40
Variance Gamma 40
Variance Gamma with Stochastic Arrival 43
Variance Gamma with Gamma Arrival Rate 45
CHAPTER 2
The Inference Problem 46
Introduction 46
Using Option Prices 49
Direction Set (Powell) Method 49
Numeric Tests 50
The Distribution of the Errors 50
Using Stock Prices 54
The Likelihood Function 54
Filtering 57
The Simple and Extended Kalman Filters 59
The Unscented Kalman Filter 62
Kushner’s Nonlinear Filter 65
Parameter Learning 67
Parameter Estimation via MLE 81
Diagnostics 95
Particle Filtering 98
Comparing Heston with Other Models 120
The Performance of the Inference Tools 127
The Bayesian Approach 144
Using the Characteristic Function 157
Introducing Jumps 158
Pure Jump Models 168
Recapitulation 184
Model Identification 185
Convergence Issues and Solutions 185
CHAPTER 3
The Consistency Problem 187
Introduction 187
The Consistency Test 189
The Setting 190

The Cross-Sectional Results 190
Robustness Issues for the Cross-Sectional Method 190
Time-Series Results 193
Financial Interpretation 194
The Peso Theory 197
Background 197
Numeric Results 199
Trading Strategies 199
Skewness Trades 200
Kurtosis Trades 200
Directional Risks 200
An Exact Replication 202
The Mirror Trades 203
An Example of the Skewness Trade 203
Multiple Trades 208
High Volatility-of-Volatility and High Correlation 209
Non-Gaussian Case 213
VGSA 215
AWord of Caution 218
Foreign Exchange, Fixed Income, and Other Markets 219
Foreign Exchange 219
Fixed Income 220
References 224
Index 236