Stochastic Finance: An Introduction in Discrete Time (De Gruyter Studies in Mathematics)
By Hans Follmer, Alexander Schied,
* Publisher: Walter de Gruyter Inc
* Number Of Pages: 400
* Publication Date: 2002-08-01
* Sales Rank: 1569705
* ISBN / ASIN: 3110171198
* EAN: 9783110171198
* Binding: Hardcover
* Manufacturer: Walter de Gruyter Inc
* Studio: Walter de Gruyter Inc
* Average Rating: 5
* Total Reviews: 1
Book Description:
This book is an introduction to financial mathematics for mathematicians. It is intended both for graduate students with a certain background in probability theory as well as for professional mathematicians in industry and academia. In contrast to many textbooks on mathematical finance, only discrete-time stochastic models are considered. This setting has the advantage that the text can concentrate from the beginning on typical problems which are suggested by financial applications. Moreover, certain principles, such as the general incompleteness of realistic market models, become thus more transparent and visible. On the other hand, all models are based on general probability spaces, and so the text captures the interplay between probability theory and functional analysis which is typical for modern mathematical finance.
The first part of the book contains a study of financial investments in a static one-period market model. Here, an investor faces intrinsic risk and uncertainty, which cannot be hedged away. The tools presented to deal with this situation range from the classical theory of expected utility until the more recent development of measures of risk.
In the second part of the book, the idea of dynamic hedging and arbitrage-free pricing of contingent claims is developed in a multi-period framework. Such market models are typically incomplete, and particular focus is given to
methods combining the dynamic hedging of a risky position with the tools of assessing risk and uncertainty as presented in part.
Contents: Mathematical finance in one period: Arbitrage theory. Expected utility. Optimal investments. Measures of risk Dynamic Arbitrage Theory: Dynamic hedging of contingent claims. American contingent claims. Optional decomposition and super-hedging. Efficient hedging in incomplete markets. Minimizing the hedging error. Hedging under constraints References. Index
Date: 2004-04-20 Rating: 5
Review:
Excellent book on mathematical finance
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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