The Volatility Surface A Practitioner's Guide

by Gatheral, Jim; Taleb, Nassim Nicholas

• ISBN13: 9780471792512
• ISBN10: 0471792519
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-09-11
• Publisher: Wiley

Summary

"Written by a practitioner for practitioners, The Volatility Surface examines why options are priced as they are and - starting from a powerful representation of implied volatility in terms of a weighted average of realized volatilities - explores the implications of various popular models for pricing."--BOOK JACKET.

Author Biography

JIM GATHERAL is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.Dr. Gatheral obtained a PhD in theoretical physics from Cambridge Universityin 1983. Since then, he has been involved in all of the major derivative product areasas a bookrunner, risk manager, and quantitative analyst in London, Tokyo, and New York. From 1997 to 2005, Dr. Gatheral headed the Equity Quantitative Analytics group at Merrill Lynch. His current research focus is equity market microstructure and algorithmic trading.

Table of Contents

List of Figures	xiii
List of Tables	xix
Foreword	xxi
Preface	xxiii
Acknowledgments	xxvii
CHAPTER 1 Stochastic Volatility and Local Volatility	1	(14)
Stochastic Volatility	1	(6)
Derivation of the Valuation Equation	4	(3)
Local Volatility	7	(8)
History	7	(1)
A Brief Review of Dupire's Work	8	(1)
Derivation of the Dupire Equation	9	(2)
Local Volatility in Terms of Implied Volatility	11	(2)
Special Case: No Skew	13	(1)
Local Variance as a Conditional Expectation of Instantaneous Variance	13	(2)
CHAPTER 2 The Heston Model	15	(10)
The Process	15	(1)
The Heston Solution for European Options	16	(4)
A Digression: The Complex Logarithm in the Integration (2.13)	19	(1)
Derivation of the Heston Characteristic Function	20	(1)
Simulation of the Heston Process	21	(4)
Milstein Discretization	22	(1)
Sampling from the Exact Transition Law	23	(1)
Why the Heston Model Is so Popular	24	(1)
CHAPTER 3 The Implied Volatility Surface	25	(18)
Getting Implied Volatility from Local Volatilities	25	(6)
Model Calibration	25	(1)
Understanding Implied Volatility	26	(5)
Local Volatility in the Heston Model	31	(2)
Ansatz	32	(1)
Implied Volatility in the Heston Model	33	(3)
The Term Structure of Black-Scholes Implied Volatility in the Heston Model	34	(1)
The Black-Scholes Implied Volatility Skew in the Heston Model	35	(1)
The SPX Implied Volatility Surface	36	(7)
Another Digression: The SVI Parameterization	37	(3)
A Heston Fit to the Data	40	(2)
Final Remarks on SV Models and Fitting the Volatility Surface	42	(1)
CHAPTER 4 The Heston-Nandi Model	43	(7)
Local Variance in the Heston-Nandi Model	43	(1)
A Numerical Example	44	(5)
The Heston-Nandi Density	45	(1)
Computation of Local Volatilities	45	(1)
Computation of Implied volatilities	46	(3)
Discussion of Results	49	(1)
CHAPTER 5 Adding Jumps	50	(24)
Why Jumps are Needed	50	(2)
Jump Diffusion	52	(4)
Derivation of the Valuation Equation	52	(2)
Uncertain Jump Size	54	(2)
Characteristic Function Methods	56	(9)
Lévy Processes	56	(1)
Examples of Characteristic Functions for Specific Processes	57	(1)
Computing Option Prices from the Characteristic Function	58	(1)
Proof of (5.6)	58	(2)
Computing Implied Volatility	60	(1)
Computing the At-the-Money Volatility Skew	60	(1)
How Jumps Impact the Volatility Skew	61	(4)
Stochastic Volatility Plus Jumps	65	(9)
Stochastic Volatility Plus Jumps in the Underlying Only (SVJ)	65	(1)
Some Empirical Fits to the SPX Volatility Surface	66	(2)
Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ)	68	(3)
SVJ Fit to the September 15, 2005, SPX Option Data	71	(2)
Why the SVJ Model Wins	73	(1)
CHAPTER 6 Modeling Default Risk	74	(13)
Merton's Model of Default	74	(3)
Intuition	75	(1)
Implications for the Volatility Skew	76	(1)
Capital Structure Arbitrage	77	(2)
Put-Call Parity	77	(1)
The Arbitrage	78	(1)
Local and Implied Volatility in the Jump-to-Ruin Model	79	(3)
The Effect of Default Risk on Option Prices	82	(2)
The CreditGrades Model	84	(3)
Model Setup	84	(1)
Survival Probability	85	(1)
Equity Volatility	86	(1)
Model Calibration	86	(1)
CHAPTER 7 Volatility Surface Asymptotics	87	(14)
Short Expirations	87	(2)
The Medvedev-Scaillet Result	89	(4)
The SABR Model	91	(2)
Including Jumps	93	(2)
Corollaries	94	(1)
Long Expirations: Fouque, Papanicolaou, and Sircar	95	(1)
Small Volatility of Volatility: Lewis	96	(1)
Extreme Strikes: Roger Lee	97	(3)
Example: Black-Scholes	99	(1)
Stochastic Volatility Models	99	(1)
Asymptotics in Summary	100	(1)
CHAPTER 8 Dynamics of the Volatility Surface	101	(6)
Dynamics of the Volatility Skew under Stochastic Volatility	101	(1)
Dynamics of the Volatility Skew under Local Volatility	102	(1)
Stochastic Implied Volatility Models	103	(1)
Digital Options and Digital Cliquets	103	(4)
Valuing Digital Options	104	(1)
Digital Cliquets	104	(3)
CHAPTER 9 Barrier Options	107	(15)
Definitions	107	(1)
Limiting Cases	108	(1)
Limit Orders	108	(1)
European Capped Calls	109	(1)
The Reflection Principle	109	(3)
The Lookback Hedging Argument	112	(1)
One-Touch Options Again	113	(1)
Put-Call Symmetry	113	(1)
QuasiStatic Hedging and Qualitative Valuation	114	(3)
Out-of-the-Money Barrier Options	114	(1)
One-Touch Options	115	(1)
Live-Out Options	116	(1)
Lookback Options	117	(1)
Adjusting for Discrete Monitoring	117	(3)
Discretely Monitored Lookback Options	119	(1)
Parisian Options	120	(1)
Some Applications of Barrier Options	120	(1)
Ladders	120	(1)
Ranges	120	(1)
Conclusion	121	(1)
CHAPTER 10 Exotic Cliquets 122
Locally Capped Globally Floored Cliquet	122	(3)
Valuation under Heston and Local Volatility Assumptions	123	(1)
Performance	124	(1)
Reverse Cliquet	125	(2)
Valuation under Heston and Local Volatility Assumptions	126	(1)
Performance	127	(1)
Napoleon	127	(6)
Valuation under Heston and Local Volatility Assumptions	128	(2)
Performance	130	(1)
Investor Motivation	130	(1)
More on Napoleons	131	(2)
CHAPTER 11 Volatility Derivatives	133	(29)
Spanning Generalized European Payoffs	133	(3)
Example: European Options	134	(1)
Example: Amortizing Options	135	(1)
The Log Contract	135	(1)
Variance and Volatility Swaps	136	(10)
Variance Swaps	137	(1)
Variance Swaps in the Heston Model	138	(1)
Dependence on Skew and Curvature	138	(2)
The Effect of Jumps	140	(3)
Volatility Swaps	143	(1)
Convexity Adjustment in the Heston Model	144	(2)
Valuing Volatility Derivatives	146	(10)
Fair Value of the Power Payoff	146	(1)
The Laplace Transform of Quadratic Variation under Zero Correlation	147	(2)
The Fair Value of Volatility under Zero Correlation	149	(2)
A Simple Lognormal Model	151	(3)
Options on Volatility: More on Model Independence	154	(2)
Listed Quadratic-Variation Based Securities	156	(5)
The VIX Index	156	(2)
VXB Futures	158	(2)
Knock-on Benefits	160	(1)
Summary	161	(1)
Postscript	162	(1)
Bibliography	163	(6)
Index	169