Hans Föllmer - Stochastic Finance: An Introduction in Discrete Time

Hans Fllmer, Alexander Schied

Stochastic Finance

De Gruyter Graduate

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Mathematics Subject Classification 2010

Primary: 60-01, 91-01, 91-02; Secondary: 46N10, 60E15, 60G40, 60G42, 91B08, 91B16, 91B30,

91B50, 91B52, 91B70, 91G10, 91G20, 91G80, 91G99.

Authors

Prof. Dr. Hans Fllmer

Humboldt-Universitt zu Berlin

Institut fr Mathematik

Unter den Linden 6

10117 Berlin

Prof. Dr. Alexander Schied

Universitt Mannheim

Lehrstuhl fr Wirtschaftsmathematik

A 5 6

68159 Mannheim

ISBN 978-3-11-046344-6

e-ISBN (PDF) 978-3-11-046345-3

e-ISBN (EPUB) 978-3-11-046346-0

Library of Congress Cataloging-in-Publication Data

A CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

2016 Walter de Gruyter GmbH, Berlin/Boston

www.degruyter.com

In this fourth edition, we have made a number of minor improvements, added some background material, and included more than two dozen new exercises. We have also addressed many comments and suggestions we received from colleagues and students during the past five years. It is a pleasure to thank Aurlien Alfonsi, Martin Barbie, Nike Blessing, Mengyu Du, Sabina Gadzhieva, Nina Gantert, Adrian Gorniak, Ria Grindel, Nils Hansen, Timo Hirscher, Marius Kaiser, Alexander Kalinin, Thomas Knispel, Mourad Lazgham, Tobias Lutz, Simon Maurer, Heiko Preu, Max von Renesse, Uwe Schmock, Qun Wang, and Ilan Zlotin. Special thanks are due to Stefan Gerhold. Finally, we thank our publisher De Gruyter for encouraging us to prepare a fourth edition and for their helpful support.

July 2016

Hans Fllmer

Alexander Schied

This third edition of our book appears in the de Gruyter graduate textbook series. We have therefore included more than one hundred exercises. Typically, we have used the book as an introductory text for two major areas, either combined into one course or in two separate courses. The first area comprises static and dynamic arbitrage theory in discrete time. The corresponding core material is provided in . Most of the exercises we have included in this edition are therefore contained in these core chapters. The other chapters of this book can be used both as complementary material for the introductory courses and as basis for special-topics courses.

In recent years, there has been an increasing awareness, both among practitioners and in academia, of the problem of model uncertainty in finance and economics, often called Knightian uncertainty ; see, e.g., [270]. In this third edition we have put more emphasis on this issue. The theory of risk measures can be seen as a case study how to deal with model uncertainty in mathematical terms. We have therefore updated we have extended the characterization of robust preferences in terms of risk measures from the coherent to the convex case. We have also included the new Sections 3.5 and 8.3 on robust variants of the classical problems of optimal portfolio choice and efficient hedging.

It is a pleasure to express our thanks to all students and colleagues whose comments have helped us to prepare this third edition, in particular to Aurlien Alfonsi, Gnter Baigger, Francesca Biagini, Julia Brettschneider, Patrick Cheridito, Samuel Drapeau, Maren Eckhoff, Karl-Theodor Eisele, Damir Filipovic, Zicheng Hong, Kostas Kardaras, Thomas Knispel, Gesine Koch, Heinz Knig, Volker Krtschmer, Christoph Khn, Michael Kupper, Mourad Lazgham, Sven Lickfeld, Mareike Massow, Irina Penner, Ernst Presman, Michael Scheutzow, Melvin Sim, Alla Slynko, Stephan Sturm, Gregor Svindland, Long Teng, Florian Werner, Wiebke Wittm, and Lei Wu. Special thanks are due to Yuliya Mishura and Georgiy Shevchenko, our translators for the Russian edition.

November 2010

Hans Fllmer

Alexander Schied

Since the publication of the first edition we have used it as the basis for several courses. These include courses for a whole semester on Mathematical Finance in Berlin and also short courses on special topics such as risk measures given at the Institut Henri Poincar in Paris, at the Department of Operations Research at Cornell University, at the Academia Sinica in Taipei, and at the 8th Symposium on Probability and Stochastic Processes in Puebla. In the process we have made a large number of minor corrections, we have discovered many opportunities for simplification and clarification, and we have also learned more about several topics. As a result, major parts of this book have been improved or even entirely rewritten. Among them are those on robust representations of risk measures, arbitrage-free pricing of contingent claims, exotic derivatives in the CRR model, convergence to BlackScholes prices, and stability under pasting with its connections to dynamically consistent coherent risk measures. In addition, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures, of concave distortions, and of the relations between risk measures and Choquet integration.

It is a pleasure to express our thanks to all students and colleagues whose comments have helped us to prepare this second edition, in particular to Dirk Becherer, Hans Bhler, Rose-Anne Dana, Ulrich Horst, Mesrop Janunts, Christoph Khn, Maren Liese, Harald Luschgy, Holger Pint, Philip Protter, Lothar Rogge, Stephan Sturm, Stefan Weber, Wiebke Wittm, and Ching-Tang Wu. Special thanks are due to Peter Bank and to Yuliya Mishura and Georgiy Shevchenko, our translators for the Russian edition. Finally, we thank Irene Zimmermann and Manfred Karbe of de Gruyter Verlag for urging us to write a second edition and for their efficient support.