Description
Book SynopsisA new, more accurate take on the classical approach to volatility evaluation Inside Volatility Filtering presents a new approach to volatility estimation, using financial econometrics based on a more accurate estimation of the hidden state.
Table of ContentsForeword ix
Acknowledgments (Second Edition) xi
Acknowledgments (First Edition) xiii
Introduction (Second Edition) xv
Introduction (First Edition) xvii
Summary xvii
Contributions and Further Research xxiii
Data and Programs xxiv
CHAPTER 1 The Volatility Problem 1
Introduction 1
The Stock Market 2
The Stock Price Process 2
Historic Volatility 3
The Derivatives Market 5
The Black-Scholes Approach 5
The Cox Ross Rubinstein Approach 7
Jump Diffusion and Level-Dependent Volatility 8
Jump Diffusion 8
Level-Dependent Volatility 11
Local Volatility 14
The Dupire Approach 14
The Derman Kani Approach 17
Stability Issues 18
Calibration Frequency 19
Stochastic Volatility 21
Stochastic Volatility Processes 21
GARCH and Diffusion Limits 22
The Pricing PDE under Stochastic Volatility 26
The Market Price of Volatility Risk 26
The Two-Factor PDE 27
The Generalized Fourier Transform 28
The Transform Technique 28
Special Cases 30
The Mixing Solution 32
The Romano Touzi Approach 32
A One-Factor Monte-Carlo Technique 34
The Long-Term Asymptotic Case 35
The Deterministic Case 35
The Stochastic Case 37
A Series Expansion on Volatility-of-Volatility 39
Local Volatility Stochastic Volatility Models 42
Stochastic Implied Volatility 43
Joint SPX and VIX Dynamics 45
Pure-Jump Models 47
Variance Gamma 47
Variance Gamma with Stochastic Arrival 51
Variance Gamma with Gamma Arrival Rate 53
CHAPTER 2 The Inference Problem 55
Introduction 55
Using Option Prices 58
Conjugate Gradient (Fletcher-Reeves-Polak-Ribiere) Method 59
Levenberg-Marquardt (LM) Method 59
Direction Set (Powell) Method 61
Numeric Tests 62
The Distribution of the Errors 65
Using Stock Prices 65
The Likelihood Function 65
Filtering 69
The Simple and Extended Kalman Filters 72
The Unscented Kalman Filter 74
Kushner’s Nonlinear Filter 77
Parameter Learning 80
Parameter Estimation via MLE 95
Diagnostics 108
Particle Filtering 111
Comparing Heston with Other Models 133
The Performance of the Inference Tools 141
The Bayesian Approach 158
Using the Characteristic Function 172
Introducing Jumps 174
Pure-Jump Models 184
Recapitulation 201
Model Identification 201
Convergence Issues and Solutions 202
CHAPTER 3 The Consistency Problem 203
Introduction 203
The Consistency Test 206
The Setting 206
The Cross-Sectional Results 206
Time-Series Results 209
Financial Interpretation 210
The “Peso” Theory 214
Background 214
Numeric Results 215
Trading Strategies 216
Skewness Trades 216
Kurtosis Trades 217
Directional Risks 217
An Exact Replication 219
The Mirror Trades 220
An Example of the Skewness Trade 220
Multiple Trades 225
High Volatility-of-Volatility and High Correlation 225
Non-Gaussian Case 230
VGSA 232
A Word of Caution 236
Foreign Exchange, Fixed Income, and Other Markets 237
Foreign Exchange 237
Fixed Income 238
CHAPTER 4 The Quality Problem 241
Introduction 241
An Exact Solution? 241
Nonlinear Filtering 242
Stochastic PDE 243
Wiener Chaos Expansion 244
First-Order WCE 247
Simulations 248
Second-Order WCE 251
Quality of Observations 251
Historic Spot Prices 252
Historic Option Prices 252
Conclusion 262
Bibliography 263
Index 279
