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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 ranging from large trading institutions to pension funds. The authors present mathematical and statistical tools that exploit the "bursty" nature of market volatility. The mathematics is introduced through examples and illustrated with simulations, and the approach described is validated and tested on market data.

The material is suitable for a one-semester course for graduate students who have been exposed to methods of stochastic modeling and arbitrage pricing theory in finance. It is easily accessible to derivatives practitioners in the inancial engineering industry.

Introduction

1 The Black-Scholes Theory of Derivative Pricing

1.1 Market Model

1.2 Derivative Contracts

1.3 Replicating Strategies

1.4 Risk-Neutral Pricing

1.5 Risk-Neutral Expectations and Partial Differential Equations

1.6 Complete Market

2 Introduction to Stochastic Volatility Models

2.1 Implied Volatility and the Smile Curve

2.2 Implied Deterministic Volatility

2.3 Stochastic Volatility Models

2.4 Derivative Pricing

2.5 Pricing with Equivalent Martingale Measures

2.6 Implied Volatility as a Function of Moneyness

2.7 Market Price of Volatility Risk and Data

2.8 Special Case: Uncorrelated Volatility

2.9 Summary and Conclusions

3 Scales in Mean-Reverting Stochastic Volatility

3.1 Scaling in Simple Models

3.2 Models of Clustering

3.3 Convergence to Black-Scholes under Fast Mean-Reverting Volatility

3.4 Scales in the Returns Process

4 Tools for Estimating the Rate of Mean Reversion

4.1 Model and Data

4.2 Variogram Analysis

4.3 Spectral Analysis

5 Asymptotics for Pricing European Derivatives

5.1 Preliminaries

5.2 The Formal Expansion

5.3 Implied Volatilities and Calibration

5.4 Accuracy of the Approximation

5.5 Region of Validity

6 Implementation and Stability

6.1 Step-by-Step Procedure

6.2 Comments about the Method

6.3 Dividends

6.4 The Second Correction

7 Hedging Strategies

8 Application to Exotic Derivatives

9 Application to American Derivatives

10 Generalizations

11 Applications to Interest-Rate Models

Index

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书籍介绍

This book, first published in 2000, 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.