de Gruyter Graduate. Berlin: de Gruyter (ISBN 978-3-11-021804-6/pbk; 978-3-11-021805-3/ebook). xi, 544 p. EUR 49.95; $ 69.00 (2011).

This book is an excellent introduction to stochastic finance in discrete time. The first part of the book contains 4 chapters. The first chapter, which is the core of this part, deals with the arbitrage theory for one period financial markets. This chapter contains many significant exercises and examples which are essential for a deep understanding of the material. Furthermore, chapter 1 provides a very good introduction for the part about dynamic arbitrage theory. Other subjects that are included in the first part of the book are preferences relations and their representations, utility maximization and equilibrium, and monetary measures of risk. The second part of the book is concerned with dynamical hedging. The main topic of this part is dynamical arbitrage theory which is explained in Chapters 5--7. In these chapters, the authors explain in detail the probabilistic approach to arbitrage theory and pricing of European and American options in general (incomplete) discrete time markets. Although the book deals with discrete time, it contains a section which explains the convergence of binomial models to the Black-Scholes model. Chapter 8 deals with efficient hedging, where the main emphasis is quantile hedging and shortfall risk minimization. This corresponds to the case where the investor may not be willing to tie in a hedging portfolio the full initial capital required for a perfect hedge. Other topics that are included in the second part are hedging under constraints, mean-variance hedging, and dynamic risk measures.
The current revised and extended edition contains many exercises, a new chapter on dynamical risk measures, and new sections on robust utility maximization and efficient hedging with convex risk measures.
The book is suitable for an audience with a background in discrete probability and functional analysis. It can be very useful for students, lecturers and researchers.