Book Description
This book is an introduction to financial mathematics. It is intended for graduate students in mathematics and for researchers working in academia and industry.
The focus on stochastic models in discrete time has two immediate benefits. First, the probabilistic machinery is simpler, and one can discuss right away some of the key problems in the theory of pricing and hedging of financial derivatives. Second, the paradigm of a complete financial market, where all derivatives admit a perfect hedge, becomes the exception rather than the rule. Thus, the need to confront the intrinsic risks arising from market incomleteness appears at a very early stage.
The first part of the book contains a study of a simple oneperiod model, which also serves as a building block for later developments. Topics include the characterization of arbitragefree markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of financial risk.
In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk.
This fourth, newly revised edition contains more than one hundred exercises. It also includes material on risk measures and the related issue of model uncertainty, in particular a chapter on dynamic risk measures and sections on robust utility maximization and on efficient hedging with convex risk measures.
Contents:
Part I: Mathematical finance in one period
Arbitrage theory
Preferences
Optimality and equilibrium
Monetary measures of risk
Part II: Dynamic hedging
Dynamic arbitrage theory
American contingent claims
Superhedging
Efficient hedging
Hedging under constraints
Minimizing the hedging error
Dynamic risk measures
Table of Contents
 Cover
 Title
 Copyright
 Preface to the fourth edition
 Preface to the third edition
 Preface to the second edition
 Preface to the first edition
 Contents

Part I: Mathematical finance in one period
 1 Arbitrage theory
 2 Preferences
 3 Optimality and equilibrium

4 Monetary measures of risk
 4.1 Risk measures and their acceptance sets
 4.2 Robust representation of convex risk measures
 4.3 Convex risk measures on L∞
 4.4 Value at Risk
 4.5 Lawinvariant risk measures
 4.6 Concave distortions
 4.7 Comonotonic risk measures
 4.8 Measures of risk in a financial market
 4.9 Utilitybased shortfall risk and divergence risk measures
 Part II: Dynamic hedging
 Appendix
 Bibliographical notes
 References
 List of symbols
 Index
Product Information
 Title: Stochastic Finance, 4th Edition
 Author(s):
 Release date: July 2016
 Publisher(s): De Gruyter
 ISBN: 9783110463460