Inside Volatility Arbitrage E-Book

Alireza Javaheri

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 289

Veröffentlichungsjahr: 2011

Beschreibung

Today?s traders want to know when volatility is a sign that the skyis falling (and they should stay out of the market), and when it isa sign of a possible trading opportunity. Inside VolatilityArbitrage can help them do this. Author and financial expertAlireza Javaheri uses the classic approach to evaluating volatility-- time series and financial econometrics -- in a way that hebelieves is superior to methods presently used by marketparticipants. He also suggests that there may be "skewness" tradingopportunities that can be used to trade the markets moreprofitably. Filled with in-depth insight and expert advice,Inside Volatility Arbitrage will help traders discover when"skewness" may present valuable trading opportunities as well aswhy it can be so profitable.

Leseprobe

Contents

Illustrations

Acknowledgments

Introduction

Chapter 1: The Volatility Problem

Introduction

The Stock Market

The Derivatives Market

Jump Diffusion and Level-Dependent Volatility

Local Volatility

Stochastic Volatility

The Pricing PDE Under Stochastic Volatility

The Generalized Fourier Transform

The Mixing Solution

The Long-Term Asymptotic Case

Pure-Jump Models

Chapter 2: The Inference Problem

Introduction

Using Option Prices

Using Stock Prices

Recapitulation

Chapter 3: The Consistency Problem

Introduction

The Consistency Test

The PESO Theory

Trading Strategies

Non-Gaussian Case

A Word of Caution

Foreign Exchange, Fixed Income, and Other Markets

References

Index

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Copyright © 2005 by Alireza Javaheri. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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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

Illustrations

Figures

1.1The SPX Historic Rolling Volatility from 2000/01/03 to 2001/12/31.1.2The SPX Volatility Smile on February 12, 2002 with Index = $1107.50, 1 Month and 7 Months to Maturity.1.3The CEV Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity.1.4The BCG Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity.1.5The GARCH Monte Carlo Simulation with the Square-Root Model for SPX on February 12, 2002 with Index = $1107.50, 1 Month to Maturity.1.6The SPX implied surface as of 03/09/2004.1.7Mixing 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.1.8Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model.1.9Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model.1.10Comparing the Volatility-of-Volatility Series Expansion with the Monte Carlo Mixing Model.1.11The Gamma Cumulative Distribution Function P(a, x) for Various Values of the Parameter a.1.12The Modified Bessel Function of Second Kind for a Given Parameter.1.13The Modified Bessel Function of Second Kind as a Function of the Parameter.2.1The S&P500 Volatility Surface as of 05/21/2002 with Index = 1079.88.2.2Mixing 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.2.3Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 09/21/2002.2.4Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 12/21/2002.2.5Mixing Monte Carlo Simulation with the Square-Root Model for SPX on 05/21/2002 with Index = $1079.88, Maturity 03/22/2003.2.6A Simple Example for the Joint Filter.2.7The EKF Estimation (Example 1) for the Drift Parameter ω.2.8The EKF Estimation (Example 1) for the Drift Parameter θ.2.9The EKF Estimation (Example 1) for the Volatility-of-Volatility Parameter ξ.2.10The EKF Estimation (Example 1) for the Correlation Parameter ρ.2.11Joint EKF Estimation for the Parameter ω.2.12Joint EKF Estimation for the Parameter θ.2.13Joint EKF Estimation for the Parameter ξ.2.14Joint EKF Estimation for the Parameter ρ.2.15Joint 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 252.2.16The SPX Historic Data (1996–2001) is Filtered via EKF and UKF.2.17The EKF and UKF Absolute Filtering Errors for the Same Time Series.2.18Histogram for Filtered Data via EKF versus the Normal Distribution.2.19Variograms for Filtered Data via EKF and UKF.2.20Variograms for Filtered Data via EKF and UKF.2.21Filtering Errors: Extended Kalman Filter and Extended Particle Filter Are Applied to the One-Dimensional Heston Model.2.22Filtering Errors: All Filters Are Applied to the One-Dimensional Heston Model.2.23Filters Are Applied to the One-Dimensional Heston Model.2.24The EKF and GHF Are Applied to the One-Dimensional Heston Model.2.25The EPF Without and with the Metropolis-Hastings Step Is Applied to the One-Dimensional Heston Model.2.26Comparison of EKF Filtering Errors for Heston, GARCH, and 3/2 Models.2.27Comparison of UKF Filtering Errors for Heston, GARCH, and 3/2 Models.2.28Comparison of EPF Filtering Errors for Heston, GARCH, and 3/2 Models.2.29Comparison of UPF Filtering Errors for Heston, GARCH, and 3/2 Models.2.30Comparison of Filtering Errors for the Heston Model.2.31Comparison of Filtering Errors for the GARCH Model.2.32Comparison of Filtering Errors for the 3/2 Model.2.33Simulated Stock Price Path via Heston Using Ψ*.2.34f(ω) = L(ω, , , ) Has a Good Slope Around = 0.10.2.35f(θ) = L(, θ, ) Has a Good Slope Around = 10.0.2.36f(ξ) = L(, , ξ, ) Is Flat Around = 0.03.2.37f(ρ) = L(, , , ρ) Is Flat and Irregular Around = −0.50.2.38f(ξ) = L(, , , ) via EKF for N = 5000 Points.2.39f(ξ) = L(, , ξ, ) via EKF for N = 50,000 Points.2.40f(ξ) = L(, , ξ, ) via EKF for N = 100,000 Points.2.41f(ξ) = L(, , ξ, ) via EKF for N = 500,000 Points.2.42Density for Estimated from 500 Paths of Length 5000 via EKF.2.43Density for Estimated from 500 Paths of Length 5000 via EKF.2.44Density for Estimated from 500 Paths of Length 5000 via EKF.2.45Density for Estimated from 500 Paths of Length 5000 via EKF.2.46Gibbs Sampler for μ in N(μ, σ).2.47Gibbs Sampler for σ in N(μ, σ).2.48Metropolis-Hastings Algorithm for μ in N(μ, σ).2.49Metropolis-Hastings Algorithm for σ in N(μ, σ).2.50Plots of the Incomplete Beta Function.2.51Comparison of EPF Results for Heston and Heston+Jumps Models. The presence of jumps can be seen in the residuals.2.52Comparison of EPF Results for Simulated and Estimated Jump-Diffusion Time Series.2.53The Simulated Arrival Rates via Ψ = (κ = 0, η = 0, λ = 0, σ = 0.2, θ = 0.02, ν = 0.005) and Ψ = (κ = 0.13, η = 0, λ = 0.40, σ = 0.2, θ = 0.02, ν = 0.005) Are Quite Different; compare with Figure 2.54.2.54However, the Simulated Log Stock Prices are Close.2.55The Observation Errors for the VGSA Model with a Generic Particle Filter.2.56The Observation Errors for the VGSA model and an Extended Particle filter.2.57The VGSA Residuals Histogram.2.58The VGSA Residuals Variogram.2.59Simulation of VGG-based Log Stock Prices with Two Different Parameter Sets Ψ = (μa = 10.0, νa = 0.01, ν = 0.05, σ = 0.2, θ = 0.002) and Ψ = (9.17, 0.19, 0.012, 0.21,0.0019).3.1Implied Volatilities of Close to ATM Puts and Calls as of 01/02/2002.3.2The Observations Have Little Sensitivity to the Volatility Parameters.3.3The state Has a Great Deal of Sensitivity to the Volatility Parameters.3.4The Observations Have a Great Deal of Sensitivity to the Drift Parameters.3.5The State Has a Great Deal of Sensitivity to the Drift Parameters.3.6Comparing SPX Cross-Sectional and Time-Series Volatility Smiles (with Historic ξ and ρ) as of January 2, 2002.3.7A Generic Example of a Skewness Strategy to Take Advantage of the Undervaluation of the Skew by Options.3.8A Generic Example of a Kurtosis Strategy to Take Advantage of the Overvaluation of the Kurtosis by Options.3.9Historic Spot Level Movements During the Trade Period.3.10Hedging PnL Generated During the Trade Period.3.11Cumulative Hedging PnL Generated During the Trade Period.3.12A Strong Option-Implied Skew: Comparing MMM (3M Co) Cross-Sectional and Time-Series Volatility Smiles as of March 28, 2003.3.13A Weak Option-Implied Skew: Comparing CMI (Cummins Inc) Cross-Sectional and Time-Series Volatility Smiles as of March 28, 2003.3.14GW (Grey Wolf Inc.) Historic Prices (03/31/2002–03/31/2003) Show a High Volatility-of-Volatility But a Weak Stock-Volatility Correlation.3.15The Historic GW (Grey Wolf Inc.) Skew Is Low and Not in Agreement with the Options Prices.3.16MSFT (Microsoft) Historic Prices (03/31/2002–03/31/2003) Show a High Volatility-of-Volatility and a Strong Negative Stock-Volatility Correlation.3.17The Historic MSFT (Microsoft) Skew Is High and in Agreement with the Options Prices.3.18NDX (Nasdaq) Historic Prices (03/31/2002–03/31/2003) Show a High Volatility-of-Volatility and a Strong Negative Stock-Volatility Correlation.3.19The Historic NDX (Nasdaq) Skew Is High and in Agreement with the Options Prices.3.20Arrival Rates for Simulated SPX Prices Using Ψ = (κ = 0.0000, η = 0.0000, λ = 0.000000, σ = 0.117200, θ = 0.0056, ν = 0.002) and Ψ = (κ = 79.499687, η = 3.557702, λ = 0.000000, σ = 0.049656, θ = 0.006801, ν = 0.008660, μ = 0.030699).3.21Gamma Times for Simulated SPX Prices Using Ψ = (κ = 0.0000, η = 0.0000, λ = 0.000000, σ = 0.117200, θ = 0.0056, ν = 0.002) and Ψ = (κ = 79.499687, η = 3.557702, λ = 0.000000, σ = 0.049656, θ = 0.006801, ν = 0.008660, μ = 0.030699).3.22Log Stock Prices for Simulated SPX Prices Using Ψ = (κ = 0.0000, η = 0.0000, λ = 0.000000, σ = 0.117200, θ = 0.0056, ν = 0.002) and Ψ = (κ = 79.499687, η = 3.557702, λ = 0.000000, σ = 0.049656, θ = 0.006801, ν = 0.008660, μ = 0.030699).3.23A Time Series of the Euro Index from January 2000 to January 2005.

Tables

1.1SPX Implied Surface as of 03/09/2004. T is the maturity and M = K/S the inverse of the moneyness.1.2Heston Prices Fitted to the 2004/03/09 Surface.2.1The Estimation is Performed for SPX on t = 05/21/2002 with Index = $1079.88 for Different Maturities T.2.2The True Parameter Set Ψ* Used for Data Simulation.2.3The Initial Parameter Set Ψ0 Used for the Optimization Process.2.4The Optimal Parameter Set Tables .2.5The Optimal EKF Parameters and Given a Sample Size N.2.6The True Parameter Set Ψ* Used for Data Generation.2.7The Initial Parameter Set Ψ0 Used for the Optimization Process.2.8The Optimal EKF Parameter Set Given a Sample Size N.2.9The Optimal EKF Parameter Set via the HRS Approximation Given a Sample Size N.2.10The Optimal PF Parameter Set Given a Sample Size N.2.11Real and Optimal Parameter Sets Obtained via NGARCH MLE.2.12Real and Optimal Parameter Sets Obtained via NGARCH MLE as well as EKF.2.13The Optimal Parameter Set for 5,000,000 Data Points.2.14Mean and (Standard Deviation) for the Estimation of Each Parameter via EKF Over P = 500 Paths of Lengths N = 5000 and N = 50,000.2.15MPE and RMSE for the VGSA Model Under a Generic PF as well as the EPF3.1Average Optimal Heston Parameter Set (Under the Risk-Neutral Distribution) Obtained via LSE Applied to One-Year SPX Options in January 2002.3.2Average Optimal Heston Parameter Set (Under the Statistical Distribution) Obtained via Filtered MLE Applied to SPX Between January 1992 and January 2004.3.3VGSA Statistical Parameters Estimated via PF.3.4VGSA Risk-Neutral Arrival-Rate Parameters Estimated from Carr et al. [48].3.5The Volatility and Correlation Parameters for the Cross-Sectional and Time-Series Approaches.

Acknowledgments

This book is based upon my Ph.D. dissertation at École des Mines de Paris. I would like to thank my advisor, Alain Galli, for his guidance and help. Many thanks go to Margaret Armstrong and Delphine Lautier and the entire CERNA team for their support.

A special thank-you goes to Yves Rouchaleau for helping make all this possible in the first place.

I would like to sincerely thank other committee members, Marco Avellaneda, Lane Hughston, Piotr Karasinski, and Bernard Lapeyre, for their comments and time.

I am grateful to Farshid Asl, Peter Carr, Raphael Douady, Robert Engle, Stephen Figlewski, Espen Haug, Ali Hirsa, Michael Johannes, Simon Julier, Alan Lewis, Dilip Madan, Vlad Piterbarg, Youssef Randjiou, David Wong, and the participants at ICBI 2003 and 2004 for all the interesting discussions and idea exchanges.

I am particularly indebted to Paul Wilmott for encouraging me to speak with Wiley about converting my dissertation into this book.

Finally, I would like to thank my wife, Firoozeh, and my daughters, Neda and Ariana, for their patience and support.

Introduction

SUMMARY

This book focuses on developing methodologies for estimating stochastic volatility (SV) parameters from the stock-price time series under a classical framework. The text contains three chapters structured as follows.

In Chapter 1, we shall introduce and discuss the concept of various parametric SV models. This chapter represents a brief survey of the existing literature on the subject of nondeterministic volatility.