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.

• Gives thorough but easy presentation of stochastic calculus for financial models • Covers all material needed for masters-level course on derivatives • Written by leading authorities in stochastic modelling

### Contents

1. The Black-Scholes theory of derivative pricing; 2. Introduction to stochastic volatility models; 3. Scales in mean-reverting stochastic volatility; 4. Tools for estimating the rate of mean-reversion; 5. Symptotics for pricing European derivatives; 6. Implementation and stability; 7. Hedging strategies; 8. Application to exotic derivatives; 9. Application to American derivatives; 10. Generalizations; 11. Applications to interest rates models.

### Reviews

'… provides a good overview to the theoretical and practical problems when dealing with stochastic volatility'. Ralf Korn, Mathematical Methods of Operations Research

'… something genuinely new … explained with admirable clarity in this extremely well-written book … [which] is short and to the point, and the production quality is high. Buy it.' Mark Davis, Risk Magazine

'… well written and makes ideal reading for a graduate course on mathematical finance. The authors took great care in making their ideas clear. I support this text strongly and recommend it for the intended audience.' P. A. L. Embrechts, Publication of the International Statistical Institute

'Thanks to a well-written first chapter on the Black-Scholes theory of derivative pricing, the book is essentially self-contained if one has some basic knowledge in stochastic methods and arbitrage pricing. Its style is largely informal which makes it also accessible to practitioners in the finance industry.' M. Schweizer, Zentralblatt für Mathematik

'… an excellent book that succeeds admirably in all its aims. It can satisfy both practitioners and researchers at the same time. It is very well written and it is concise and informative.' Angelos Dassios, The Statistician

'I consider this book to be an outstanding achievement. the theory is practically very relevant and scientifically on a high level. The book also serves as a good introduction into the basic ideas of Mathematical Finance, putting emphasis on the techniques of partial differential equations. It can therefore also be recommended to readers with little knowledge about Mathematical Finance.' Monatshefte für Mathematik