Derivatives in Financial Markets With Stochastic Volatility

by Jean-Pierre Fouque , George Papanicolaou , K. Ronnie Sircar

• ISBN13: 9780521791632
• ISBN10: 0521791634
• Format: Hardcover
• Copyright: 2000-07-03
• Publisher: Cambridge University Press

Summary

This book addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These problems are important to investors from large trading institutions to pension funds. It presents mathematical and statistical tools that exploit the bursty nature of market volatility. The mathematics is introduced through examples and illustrated with simulations and the modeling approach that is described is validated and tested on market data. The material is suitable for a one semester course for graduate students who have had exposure to methods of stochastic modeling and arbitrage pricing theory in finance. It is easily accessible to derivatives practitioners in the financial engineering industry.

Table of Contents

Introduction	xi
The Black---Scholes Theory of Derivative Pricing	1	(32)
Market Model	1	(7)
Brownian Motion	2	(1)
Stochastic Integrals	3	(1)
Risky Asset Price Model	4	(2)
Ito's Formula	6	(2)
Lognormal Risky Asset Price	8	(1)
Derivative Contracts	8	(4)
European Call and Put Options	9	(1)
American Options	10	(1)
Other Exotic Options	11	(1)
Replicating Strategies	12	(6)
Replicating Self-Financing Portfolios	12	(1)
The Black--Scholes Partial Differential Equation	13	(2)
Pricing to Hedge	15	(1)
The Black---Scholes Formula	15	(3)
Risk-Neutral Pricing	18	(5)
Equivalent Martingale Measure	19	(1)
Self-Financing Portfolios	20	(1)
Risk-Neutral Valuation	21	(1)
Using the Markov Property	22	(1)
Risk-Neutral Expectations and Partial Differential Equations	23	(8)
Infinitesimal Generators and Associated Martingales	24	(1)
Conditional Expectations and Parabolic Partial Differential Equations	25	(1)
Application to the Black---Scholes Partial Differential Equation	26	(1)
American Options and Free Boundary Problems	27	(2)
Path-Dependent Derivatives	29	(2)
Complete Market	31	(2)
Introduction to Stochastic Volatility Models	33	(25)
Implied Volatility and the Smile Curve	34	(4)
Interpretation of the Smile Curve	35	(2)
What Data to Use	37	(1)
Implied Deterministic Volatility	38	(2)
Time-Dependent Volatility	38	(1)
Level-Dependent Volatility	39	(1)
Short-Time Tight Fit versus Long-Time Rough Fit	39	(1)
Stochastic Volatility Models	40	(2)
Mean-Reverting Stochastic Volatility Models	40	(2)
Stock-Price Distribution under Stochastic Volatility	42	(1)
Derivative Pricing	42	(4)
Pricing with Equivalent Martingale Measures	46	(2)
Implied Volatility as a Function of Moneyness	48	(1)
Market Price of Volatility Risk and Data	49	(2)
Special Case: Uncorrelated Volatility	51	(4)
Hull---White Formula	51	(1)
Stochastic Volatility Implies Smile	51	(2)
Remark on Correlated Volatility	53	(2)
Summary and Conclusions	55	(3)
Scales in Mean-Reverting Stochastic Volatility	58	(19)
Scaling in Simple Models	58	(1)
Models of Clustering	59	(11)
Example: Markov Chain	60	(4)
Example: Another Jump Process	64	(3)
Example: Ornstein---Uhlenbeck Process	67	(3)
Summary	70	(1)
Convergence to Black---Scholes under Fast Mean-Reverting Volatility	70	(1)
Scales in the Returns Process	71	(6)
The Returns Process	72	(1)
Returns Process with Jump Volatility	72	(1)
Returns Process with OU Volatility	73	(2)
S&P 500 Returns Process	75	(2)
Tools for Estimating the Rate of Mean Reversion	77	(10)
Model and Data	77	(2)
Mean-Reverting Stochastic Volatility	77	(1)
Discrete Data	78	(1)
Variogram Analysis	79	(5)
Computation of the Variogram	79	(1)
Comparison and Sensitivity Analysis with Simulated Data	80	(3)
The Day Effect	83	(1)
Spectral Analysis	84	(3)
Asymptotics for Pricing European Derivatives	87	(21)
Preliminaries	87	(3)
The Rescaled Stochastic Volatility Model	88	(1)
The Rescaled Pricing Equation	88	(1)
The Operator Notation	89	(1)
The Formal Expansion	90	(9)
The Diverging Terms	90	(1)
Poisson Equations	91	(2)
The Zero-Order Term	93	(1)
The First Correction	94	(2)
Universal Market Group Parameters	96	(1)
Probabilistic Interpretation of the Source Term	97	(1)
Put---Call Parity	98	(1)
The Skew Effect	98	(1)
Implied Volatilities and Calibration	99	(3)
Accuracy of the Approximation	102	(2)
Region of Validity	104	(4)
Implementation and Stability	108	(7)
Step-by-Step Procedure	108	(1)
Comments about the Method	109	(3)
Dividends	112	(1)
The Second Correction	113	(2)
Hedging Strategies	115	(9)
Black---Scholes Delta Hedging	115	(4)
The Strategy and Its Cost	116	(1)
Averaging Effect	117	(2)
Mean Self-Financing Hedging Strategy	119	(2)
Staying Close to the Price	121	(3)
Application to Exotic Derivatives	124	(8)
European Binary Options	124	(1)
Barrier Options	125	(4)
Asian Options	129	(3)
Application to American Derivatives	132	(13)
American Problem under Stochastic Volatility	132	(1)
Stochastic Volatility Correction for an American Put	133	(8)
Expansions	134	(2)
First Approximation	136	(2)
The Stochastic Volatility Correction	138	(2)
Uncorrelated Volatility	140	(1)
Probabilistic Representation	141	(1)
Numerical Computation	141	(4)
Solution of the Black---Scholes Problem	141	(1)
Computation of the Correction	142	(3)
Generalizations	145	(29)
Portfolio Optimization under Stochastic Volatility	145	(8)
Constant Volatility Merton Problem	145	(2)
Stochastic Volatility Merton Problem	147	(4)
A Practical Solution	151	(2)
Periodic Day Effect	153	(3)
Other Markovian Volatility Models	156	(3)
Markovian Jump Volatility Models	156	(2)
Pricing and Asymptotics	158	(1)
Martingale Approach	159	(6)
Main Argument	159	(2)
Decomposition Result	161	(3)
Comparison with the PDE Approach	164	(1)
Non-Markovian Models of Volatility	165	(4)
Setting: An Example	165	(1)
Asymptotics in the Non-Markovian Case	166	(3)
Multidimensional Models	169	(5)
Applications to Interest-Rate Models	174	(21)
Bond Pricing in the Vasicek Model	174	(8)
Review of the Constant Volatility Vasicek Model	174	(3)
Stochastic Volatility Vasicek Models	177	(5)
Bond Option Pricing	182	(8)
The Constant Volatility Case	183	(2)
Correction for Stochastic Volatility	185	(4)
Implications	189	(1)
Asymptotics around the CIR Model	190	(2)
Illustration from Data	192	(3)
Variogram Analysis	192	(1)
Yield Curve Fitting	193	(2)
Bibliography	195	(4)
Index	199