Table of contents for The volatility surface : a practitioner's guide / by Jim Gatheral ; foreword by Nassim Nicholas Taleb.

Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.


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Contents
Preface 13
1 Stochastic Volatility and Local Volatility 21
Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Derivation of the Valuation Equation . . . . . . . . . . . . . 24
Local Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A Brief Review of Dupire?s Work . . . . . . . . . . . . . . . 28
Derivation of the Dupire Equation . . . . . . . . . . . . . . 29
Local volatility in terms of implied volatility . . . . . . . . . 30
Special Case: No Skew . . . . . . . . . . . . . . . . . . . . . 32
Local Variance as a Conditional Expectation of Instantaneous
Variance . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 The Heston Model 35
The process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
The Heston solution for European options . . . . . . . . . . . . . 36
Derivation of the Heston Characteristic Function . . . . . . . . . 39
Simulation of the Heston process . . . . . . . . . . . . . . . . . . 40
Why the Heston model is so popular . . . . . . . . . . . . . 43
3 The Implied Volatility Surface 45
Getting Implied Volatility from Local Volatilities . . . . . . . . . . 45
Model Calibration . . . . . . . . . . . . . . . . . . . . . . . 45
Understanding Implied Volatility . . . . . . . . . . . . . . . 46
Local volatility in the Heston model . . . . . . . . . . . . . . . . . 51
Implied volatility in the Heston model . . . . . . . . . . . . . . . 53
The SPX implied volatility surface . . . . . . . . . . . . . . . . . . 55
A Heston fit to the data . . . . . . . . . . . . . . . . . . . . 59
Final remarks on SV models and fitting the volatility surface 60
9
10 INDEX
4 The Heston-Nandi Model 63
Local variance in the Heston-Nandi model . . . . . . . . . . . . . 63
A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . 64
The Heston-Nandi density . . . . . . . . . . . . . . . . . . . 64
Computation of local volatilities . . . . . . . . . . . . . . . . 65
Computation of implied volatilities . . . . . . . . . . . . . . 67
Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Adding Jumps 71
Why Jumps are Needed . . . . . . . . . . . . . . . . . . . . . . . 71
Jump diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Derivation of the valuation equation . . . . . . . . . . . . . 73
Characteristic Function Methods . . . . . . . . . . . . . . . . . . 76
L?evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . 77
Examples of characteristic functions for specific processes . . 77
Computing option prices from the characteristic function . . 79
Computing Implied Volatility . . . . . . . . . . . . . . . . . 81
Computing the At-the-money Volatility Skew . . . . . . . . . 81
How jumps impact the volatility skew . . . . . . . . . . . . 82
Stochastic Volatility plus Jumps . . . . . . . . . . . . . . . . . . . 86
Stochastic Volatility plus Jumps in the Underlying Only (SVJ) 86
Some Empirical Fits to the SPX Volatility Surface . . . . . . 87
Stochastic volatility with Simultaneous Jumps in Stock Price
and Volatility (SVJJ) . . . . . . . . . . . . . . . . . . 89
SVJ fit to the September 15, 2005 SPX option data . . . . . 91
Why the SVJ model wins . . . . . . . . . . . . . . . . . . . . 93
6 Modeling Default Risk 95
Merton?s Model of Default . . . . . . . . . . . . . . . . . . . . . . 95
Capital structure arbitrage . . . . . . . . . . . . . . . . . . . . . . 97
Local and implied volatility in the jump-to-ruin model . . . . . . . 100
The effect of default risk on option prices . . . . . . . . . . . . . . 102
The CreditGrades model . . . . . . . . . . . . . . . . . . . . . . . 104
Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Survival probability . . . . . . . . . . . . . . . . . . . . . . . 105
Equity volatility . . . . . . . . . . . . . . . . . . . . . . . . . 106
Model calibration . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Volatility Surface Asymptotics 109
Short Expirations . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
The Medvedev-Scaillet Result . . . . . . . . . . . . . . . . . . . . 111
The SABR model . . . . . . . . . . . . . . . . . . . . . . . . 113
CONTENTS 11
Including Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Long Expirations: Fouque, Papanicolaou and Sircar . . . . . . . . 116
Small Volatility of Volatility: Lewis . . . . . . . . . . . . . . . . . 117
Extreme Strikes: Roger Lee . . . . . . . . . . . . . . . . . . . . . . 118
Asymptotics in Summary . . . . . . . . . . . . . . . . . . . . . . . 120
8 Dynamics of the Volatility Surface 121
Dynamics of the Volatility Skew under Stochastic Volatility . . . . 121
Dynamics of the Volatility Skew under Local Volatility . . . . . . 122
Stochastic Implied Volatility Models . . . . . . . . . . . . . . . . 123
Digital Options and Digital Cliquets . . . . . . . . . . . . . . . . 123
Valuing Digital Options . . . . . . . . . . . . . . . . . . . . 123
Digital Cliquets . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 Barrier Options 127
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Limit Orders . . . . . . . . . . . . . . . . . . . . . . . . . . 128
European Capped Calls . . . . . . . . . . . . . . . . . . . . 129
The Reflection Principle . . . . . . . . . . . . . . . . . . . . . . . 129
The Lookback Hedging Argument . . . . . . . . . . . . . . . . . . 132
Put-Call Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Quasi-static hedging and qualitative valuation . . . . . . . . . . . 134
Adjusting for Discrete Monitoring . . . . . . . . . . . . . . . . . . 137
Parisian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Some Applications of Barrier Options . . . . . . . . . . . . . . . . 140
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10 Exotic Cliquets 143
Locally Capped Globally Floored Cliquet . . . . . . . . . . . . . . 143
Valuation under Heston and local volatility assumptions . . 144
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Reverse Cliquet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Valuation under Heston and local volatility assumptions . . 147
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Napoleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Valuation under Heston and local volatility assumptions . . 149
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Investor motivation . . . . . . . . . . . . . . . . . . . . . . . 151
More on Napoleons . . . . . . . . . . . . . . . . . . . . . . 152
11 Volatility derivatives 155
12 INDEX
Spanning Generalized European Payoffs . . . . . . . . . . . . . . 155
The Log Contract . . . . . . . . . . . . . . . . . . . . . . . . 157
Variance and Volatility Swaps . . . . . . . . . . . . . . . . . . . . 158
Variance Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 158
Variance Swaps in the Heston Model . . . . . . . . . . . . . 159
Dependence on Skew and Curvature . . . . . . . . . . . . . 160
The Effect of Jumps . . . . . . . . . . . . . . . . . . . . . . . 162
Volatility Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 164
Convexity Adjustment in the Heston Model . . . . . . . . . 165
Valuing volatility derivatives . . . . . . . . . . . . . . . . . . . . . 167
Fair value of the power payoff . . . . . . . . . . . . . . . . . 167
The Laplace transform of quadratic variation under zero correlation
. . . . . . . . . . . . . . . . . . . . . . . . . 168
The fair value of volatility under zero correlation . . . . . . 170
A simple lognormal model . . . . . . . . . . . . . . . . . . . 172
Options on volatility: More on model-independence . . . . . 174
Listed quadratic-variation based securities . . . . . . . . . . . . . 176
The VIX index . . . . . . . . . . . . . . . . . . . . . . . . . 176
VXB futures . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Knock-on benefits . . . . . . . . . . . . . . . . . . . . . . . . 180
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Library of Congress Subject Headings for this publication:

Options (Finance) -- Prices -- Mathematical models.
Stocks -- Prices -- Mathematical models.