Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

by Jean-Pierre Fouque , George Papanicolaou , Ronnie Sircar , Knut Sølna

• ISBN13: 9780521843584
• ISBN10: 0521843588
• Format: Hardcover
• Copyright: 2011-11-21
• Publisher: Cambridge University Press

Summary

This book builds on previous and current research by the four authors, including results introduced in the book Derivatives in Financial Markets with Stochastic Volatility. Recent research demonstrates that the introduction of two time scales in volatility, a fast and a slow, is needed and efficient for capturing the main features of the observed term structure of implied volatility. For practitioners, the modeling of the implied volatility consistent with no-arbitrage is crucial. The authors present an approach to this problem which consists in combining singular and regular perturbation techniques. The book will serve a dual purpose: present 'off the shelf' formulas and calibration tools for practitioners, and introduce, explain and develop the mathematical framework to handle the multi-scale asymptotics. Detailed presentation of the analysis as well as a thorough insight into the modeling approach makes this an excellent text for a second level graduate course in financial and applied mathematics.

Author Biography

Jean-Pierre Fouque studied at the University Pierre et Marie Curie in Paris. He held positions at the French CNRS and Ecole Polytechnique, and at North Carolina State University. Since 2006, he is Professor and Director of the Center for Research in Financial Mathematics and Statistics at the University of California, Santa Barbara. George Papanicolaou was Professor of Mathematics at the Courant Institute before coming to Stanford University in 1993. He is now Robert Grimmett Professor in the Department of Mathematics at Stanford Ronnie Sircar is a Professor in the Operations Research and Financial Engineering department at Princeton University, and an affiliate member of the Bendheim Center for Finance and the Program in Applied and Computational Mathematics. Knut Slna is a Professor in the Department of Mathematics at the University of California at Irvine. He received his undergraduate and Master's degrees from the Norwegian University of Science and Technology, and his doctorate from Stanford University. He was an instructor at the Department of Mathematics, University of Utah before coming to Irvine.

Table of Contents

Introduction	p. xi
The Black-Scholes Theory of Derivative Pricing	p. 1
Market Model	p. 1
Derivative Contracts	p. 10
Replicating Strategies	p. 13
Risk-Neutral Pricing	p. 21
Risk-Neutral Expectations and Partial Differential Equations	p. 27
American Options and Free Boundary Problems	p. 32
Path-Dependent Derivatives	p. 36
First-Passage Structural Approach to Default	p. 43
Multidimensional Stochastic Calculus	p. 46
Complete Market	p. 49
Introduction to Stochastic Volatility Models	p. 51
Implied Volatility Surface	p. 52
Local Volatility	p. 57
Stochastic Volatility Models	p. 62
Derivative Pricing	p. 65
General Results on Stochastic Volatility Models	p. 74
Summary and Conclusions	p. 83
Volatility Time Scales	p. 86
A Simple Picture of Fast and Slow Time Scales	p. 86
Ergodicity and Mean-Reversion	p. 88
Examples of Mean-Reverting Processes	p. 95
Time Scales in Synthetic Returns Data	p. 110
Time Scales in Market Data	p. 114
Multiscale Models	p. 118
First-Order Perturbation Theory	p. 121
Option Pricing under Multiscale Stochastic Volatility	p. 121
Formal Regular and Singular Perturbation Analysis	p. 125
Parameter Reduction	p. 135
First-Order Approximation: Summary and Discussion	p. 137
Accuracy of First-Order Approximation	p. 138
Implied Volatility Formulas and Calibration	p. 148
Approximate Call Prices and Implied Volatilities	p. 149
Calibration Procedure	p. 154
Illustration with S&P 500 Data	p. 155
Maturity Cycles	p. 163
Higher-Order Corrections	p. 174
Application to Exotic Derivatives	p. 179
European Binary Options	p. 179
Barrier Options	p. 181
Asian Options	p. 185
Application to American Derivatives	p. 189
American Options Valuation under Stochastic Volatility	p. 189
Stochastic Volatility Correction for American Put	p. 190
Parameter Reduction	p. 195
Summary	p. 197
Hedging Strategies	p. 199
Black-Scholes Delta Hedging	p. 200
The Strategy and its Cost	p. 200
Mean Self-Financing Hedging Strategy	p. 206
A Strategy with Frozen Parameters	p. 209
Strategies Based on Implied Volatilities	p. 217
Martingale Approach to Pricing	p. 220
Non-Markovian Models of Volatility	p. 226
Extensions	p. 232
Dividends and Varying Interest Rates	p. 232
Probabilistic Representation of the Approximate Prices	p. 237
Second-Order Correction from Fast Scale	p. 238
Second-Order Corrections from Slow and Fast Scales	p. 247
Periodic Day Effect	p. 249
Markovian Jump Volatility Models	p. 251
Multidimensional Models	p. 254
Around the Heston Model	p. 259
The Heston Model	p. 260
Approximations to the Heston Model	p. 265
A Fast Mean-Reverting Correction to the Heston Model	p. 271
Large Deviations and Short Maturity Asymptotics	p. 276
Other Applications	p. 283
Application to Variance Reduction in Monte Carlo Computations	p. 283
Portfolio Optimization under Stochastic Volatility	p. 287
Application to CAPM Forward-Looking Beta Estimation	p. 296
Interest Rate Models	p. 307
The Vasicek Model	p. 307
The Bond Price and its Expansion	p. 315
The Quadratic Model	p. 327
The CIR Model	p. 330
Options on Bonds	p. 335
Structural Models with Stochastic Volatility	p. 342
Single-Name Credit Derivatives	p. 342
Multiname Credit Derivatives	p. 353
Multiscale Intensity-Based Models	p. 377
Background on Stochastic Intensity Models	p. 377
Multiname Credit Derivatives	p. 385
Symmetric Vasicek Model	p. 388
Homogeneous Group Structure	p. 402
Epilogue	p. 424
References	p. 430
Index	p. 439
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