Introduction

Summary

This book is about pricing and hedging financial derivatives under stochastic volatility in equity, interest rate, and credit markets. We demonstrate that the introduction of two time scales in volatility, a fast and a slow, is needed and is efficient for capturing the main features of the observed term structures of implied volatility, yields, or credit spreads. The present book builds on and replaces our previous book, Derivatives in Financial Markets with Stochastic Volatility, published by Cambridge University Press in 2000.

We present an approach to derivatives valuation and hedging which consists of integrating singular and regular perturbation techniques in the context of stochastic volatility. The book has a dual purpose: to present “off-the-shelf” formulas and calibration tools, and to introduce, explain, and develop the mathematical framework to handle the multiscale asymptotics.

There are many books on financial mathematics (mostly for introductory courses at the level of the Black–Scholes model). Primarily, these books deal with the case of constant volatilities, be it for stock prices, interest rates, or default intensities. This book is about analyzing these models in the presence of stochastic volatility using the powerful tools of perturbation methods. The book can be used for a second-level graduate course in Financial and Applied Mathematics.

Our goal is to address the following fundamental problem in pricing and hedging derivatives: how can traded call and put options, quoted in terms of implied volatilities, be used to price and hedge more complicated contracts?