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Inside Volatility Arbitrage Inside Volatility Arbitrage The Secrets of Skewness ALIREZA JAVAHERI John Wiley & Sons, Inc. Inside Volatility Arbitrage Founded in 1807, John Wiley & Sons is the oldest independent publish- ing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and market- ing print and electronic products and services for our customers’ professional and personal knowledge and understanding. The Wiley Finance series contains books written specifically for finance and investment professionals as well as sophisticated individual investors and their financial advisors. Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation and financial instrument analysis, as well as much more. For a list of available titles, visit our Web site at www.WileyFinance.com. Inside Volatility Arbitrage The Secrets of Skewness ALIREZA JAVAHERI John Wiley & Sons, Inc. Copyright © 2005 by Alireza Javaheri. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data Javaheri, Alireza. Inside volatility arbitrage : the secrets of skewness / Alireza Javaheri. p. cm. Includes bibliographical references and index. ISBN 0-471-73387-3 (cloth) 1. Stocks–Proces–Mathematical models. 2. Stochastic processes. I. Title. HG4636.J38 2005 332.63’222’0151922–dc22 2005004696 Printed in the United States of America 10987654321 Contents Illustrations ix Acknowledgments xv Introduction 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 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 v vi CONTENTS 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 Contents vii 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 Illustrations Figures 1.1 The SPX Historic Rolling Volatility from 2000/01/03 to 2001/12/31. 4 1.2 The SPX Volatility Smile on February 12, 2002 with Index = $1107.50, 1 Month and 7 Months to Maturity. 8 1.3 The CEV Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity. 11 1.4 The BCG Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity. 12 1.5 The GARCH Monte Carlo Simulation with the Square- Root Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity. 24 1.6 The SPX implied surface as of 03/09/2004. 31 1.7 Mixing Monte Carlo Simulation with the Square-Root Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month and 7 Months to Maturity. 33 1.8 Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model. 38 1.9 Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model. 39 1.10 Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model. 39 1.11 The Gamma Cumulative Distribution Function P(ax)for Various Values of the Parameter a.42 1.12 The Modified Bessel Function of Second Kind for a Given Parameter. 42 1.13 The Modified Bessel Function of Second Kind as a Function of the Parameter. 43 2.1 The S&P500 Volatility Surface as of 05/21/2002 with Index = 1079.88.51 2.2 Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 08/17/2002 Powell (direction set) optimization method was used for least-square calibration. 51 ix x ILLUSTRATIONS 2.3 Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 09/21/2002. 52 2.4 Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 12/21/2002. 52 2.5 Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 03/22/2003. 53 2.6 A Simple Example for the Joint Filter. 69 2.7 The EKF Estimation (Example 1) for the Drift Parameter ω.71 2.8 The EKF Estimation (Example 1) for the Drift Parameter θ.72 2.9 The EKF Estimation (Example 1) for the Volatility- of-Volatility Parameter ξ.72 2.10 The EKF Estimation (Example 1) for the Correlation Parameter ρ.73 2.11 Joint EKF Estimation for the Parameter ω.78 2.12 Joint EKF Estimation for the Parameter θ.79 2.13 Joint EKF Estimation for the Parameter ξ.79 2.14 Joint EKF Estimation for the Parameter ρ.80 2.15 Joint EKF Estimation for the Parameter ω Applied to the Heston Model as Well as to a Modified Model Where the Noise Is Reduced by a Factor 252.81 2.16 The SPX Historic Data (1996–2001) is Filtered via EKF and UKF. 84 2.17 The EKF and UKF Absolute Filtering Errors for the Same Time Series. 85 2.18 Histogram for Filtered Data via EKF versus the Normal Distribution. 86 2.19 Variograms for Filtered Data via EKF and UKF. 97 2.20 Variograms for Filtered Data via EKF and UKF. 98 2.21 Filtering Errors: Extended Kalman Filter and Extended Par- ticle Filter Are Applied to the One-Dimensional Heston Model. 115 2.22 Filtering Errors: All Filters Are Applied to the One- Dimensional Heston Model. 116 2.23 Filters Are Applied to the One-Dimensional Heston Model. 117 2.24 The EKF and GHF Are Applied to the One-Dimensional Heston Model. 118 2.25 The EPF Without and with the Metropolis-Hastings Step Is Applied to the One-Dimensional Heston Model.
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