Inside Volatility Filtering: Secrets of the Skew

"While e–trading typically starts with cash instruments and vanilla securities, it is inevitable that it will eventually encompass trading activities that lean heavily on quantitative elements such as volatility trading. As a result, the Second Edition of this book serves its intended audience well, providing an up–to–date, comprehensive review of the application of filtering techniques to volatility forecasting. While the title of each chapter is framed as a problem, the contents of each chapter represent our best guess at the answer. Employing the advances that econometricians have made in the past quarter century, the fraction of variance explained is a truly impressive accomplishment."
From the Foreword by Peter Carr, Global Head of Market Modeling, Morgan Stanley; and Executive Director, Masters in Math Finance Program, Courant Institute, New York University

The New, More Accurate Take on the Classical Approach to Volatility Evaluation

Inside Volatility Filtering, Second Edition presents a new approach to volatility estimation identifying financial econometrics based on a more accurate estimation of the hidden state. Based on the idea of "filtering," this practical guide lays out a two–step framework involving a Chapman–Kolmogorov prior distribution followed by Bayesian posterior distribution to develop a robust estimation based on all available information. This new edition gives you an edge by showing you how to:

Base volatility estimations on more accurate data Integrate past observation with Bayesian probability Exploit posterior distribution of the hidden state for optimal estimation Boost trade profitability by identifying "skewness" opportunities

Foreword 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

ALIREZA JAVAHERI is the head of Equities Quantitative Research Americas at JP Morgan and an adjunct professor of Mathematical Finance at the Courant Institute of New York University, as well as Baruch College. He has worked in the field of derivatives quantitative research since 1994 in a variety of investment banks, including Goldman Sachs and Citigroup.