In this text, the author discusses the main aspects of mathematical finance. These include arbitrage, hedging and pricing of contingent claims, portfolio optimization, incomplete and/or constrained markets, equilibrium, and transaction costs. The book outlines advances made possible during the last fifteen years due to the methodologies of stochastic analysis and control. Readers are presented with current research, and open problems are suggested. This tutorial survey of the rapidly expanding field of mathematical finance is addressed primarily to graduate students in mathematics. Familiarity is assumed with stochastic analysis and parabolic partial differential equations. The text makes significant use of students' mathematical skills, but always in connection with interesting applied problems. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students in mathematics, statistics, physics, and engineering who want to learn about the Mathematics of Finance. Reviews "Readers are presented with current research, and open problems are suggested. This tutorial survey of the rapidly expanding field of mathematical finance is addressed primarily to graduate students in mathematics. Familiarity is assumed with stochastic analysis and parabolic partial differential equations. The text makes significant use of students' mathematical skills, but always in connection with interesting applied problems."  Zentralblatt für Didaktik der Mathematik "Provides an excellent introduction to a wide range of topics in mathematical finance."  Mathematical Reviews "The young researcher/postgraduate student will be able to glance at the forefront of current research in mathematical finance. The author's clear and careful writing makes reading a pleasure. A lot of material, hitherto available only in research papers, will now reach a wider audience. This is a most useful addition to the fast growing literature on mathematical finance."  Short Book Reviews, a publication of the International Statistical Institute Table of Contents Part 1. Complete markets  Pricing
 Optimization
 Equilibrium
Part 2. Incomplete markets  Hedging
 Optimization
 Pricing
 Transaction costs
 Appendix A. Historical Notes
 Bibliography
