The Mathematics of Arbitrage
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Editorial Board M. Avellaneda G. Barone-Adesi M. Broadie M.H.A. Davis E. Derman C. Klüppelberg E. Kopp W. Schachermayer
Freddy Delbaen Walter Schachermayer •
The Mathematics of Arbitrage
Freddy Delbaen ETH Zürich Departement Mathematik Rämistr. 101 8092 Zürich Switzerland [email protected]
Walter Schachermayer Technische Universität Wien Finanz- und Versicherungsmathematik Wiedner Hauptstr. 8-10 1040 Wien Austria [email protected]
ISBN 978-3-540-21992-7
e-ISBN 978-3-540-31299-4
DOI 10.1007/978-3-540-31299-4 Library of Congress Control Number: 2005937005 Mathematics Subject Classification (2000): M13062, M27004, M12066 JEL Classification: G12, G13 Corrected 2nd Printing 2008 c Springer-Verlag Berlin Heidelberg 2006 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
To Rita and Christine with love
Preface
In 1973 F. Black and M. Scholes published their pathbreaking paper [BS 73] on option pricing. The key idea — attributed to R. Merton in a footnote of the Black-Scholes paper — is the use of trading in continuous time and the notion of arbitrage. The simple and economically very convincing “principle of noarbitrage” allows one to derive, in certain mathematical models of financial markets (such as the Samuelson model, [S 65], nowadays also referred to as the “Black-Scholes” model, based on geometric Brownian motion), unique prices for options and other contingent claims. This remarkable achievement by F. Black, M. Scholes and R. Merton had a profound effect on financial markets and it shifted the paradigm of dealing with financial risks towards the use of quite sophisticated mathematical models. It was in the late seventies that the central role of no-arbitrage arguments was crystallised in three seminal papers by M. Harrison, D. Kreps and S. Pliska ([HK 79], [HP 81], [K 81]) They considered a general framework, which allows a systematic study of different models of financial markets. The Black-Scholes model is just one, obviously very important, example embedded into the framework of a general theory. A basic insight of these papers was the intimate relation between no-arbitrage arguments on one hand, and martinga
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