Chapman & Hall/CRC Financial Mathematics Series. Boca Raton, FL: Chapman & Hall/CRC (ISBN 978-1-58488-626-6/hbk). 253 p. £ 34.99; $ 69.95 (2008).

This book is the second English edition (from 2008) of the textbook whose first French edition was among the first books that provided an excellent mathematical introduction into the theory of mathematical finance in continuous time. Aimed at a mathematically trained readership, it is still one of the best introductions to this field. The earlier parts of the text contain a brief introduction to the mathematical theory of stochastic calculus, as needed for solving mathematical problems from finance. The latter area is the main theme of the book and is covered in a wider scope -- ranging from optimal stopping problems for American options and the Black-Scholes model, over martingale and PDE methods for option pricing, to models for interest rate and credit derivatives. Also Monte-Carlo simulation methods are treated. Despite this range, the book is comparably compact, the presentation being concise and to the point. Results and definitions are rigorously stated, and references are provided for results whose proofs are omitted. Some results are provided through guided exercises, for instance the Dupire local volatility model or the caplet pricing formula for the Libor market model. This serves mathematically advanced readers well who want to learn about mathematical finance but do not need to be hand-waved through the ideas of stochastic calculus in some more informal way. On the other hand, the focus on applications sets the book apart from other mathematical books that mainly concentrate on stochastic theory with some excursions to financial applications. The new second edition has been complemented by additional topics, for instance the Libor market model, credit risk models and the change of numeraire technique. Also further exercises have been added, this includes computer experiments with code provided online in Scilab, which is open source software.