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(More customer reviews)It is well-known that the mathematical study of finance has, over the last two decades, led to a number of discoveries in stochastic analysis whose import extends beyond the boundaries of finance to other areas of mathematics. There are, currently, many good text-books which treat the mathematics of financial markets (e.g. Pliska, Bingham&Kiesel, Elliott&Kopp, Musiela&Rutkowski, Karatzas&Shreve, roughly in increasing order of difficulty. Pliska's text works only in the discrete-time framework, whereas the others move quickly to continuous-time). The text by Follmer and Schied deals only with the discrete-time case, but covers a large amount of material which you won't find in any of the other books: A thorough introduction to utility theory, excellent coverage of coherent and convex risk measures, and various approaches to hedging in incomplete markets. Each chapter quickly brings the reader close to the frontiers of research. Future research in these areas also promises to overflow the boundaries, providing new applications to other branches of functional analysis.
A word of caution: Though the text restricts itself to the "simpler" discrete-time case, thus avoiding stochastic integration, it nevertheless demands a solid background in analysis, including graduate level probability theory and functional analysis. Though not technically a requirement, some background in mathematical finance is necessary in order to understand what this book is about.
In conclusion, therefore, don't make this your first book on mathematical finance -- get Bingham&Kiesel instead. But if you have the mathematical background, and are analytically inclined, do buy it. This book is a phenomenal achievement.
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This book is an introduction to financial mathematics. The first part of the book studies a simple one-period model which serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of risk. In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Such models are typically incomplete: They involve intrinsic risks which cannot be hedged away completely. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk. In addition to many corrections and improvements, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures and of the connections between American options, superhedging, and dynamic risk measures.
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