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option_pricing_in_fractional_brownian_markets-2009(2)

option_pricing_in_fractional_brownian_markets-2009(2) -...

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Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, H. Dawid, K. Inderfurth, W. Kürsten 622
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123 Stefan Rostek Option Pricing in Fractional Brown i an Markets
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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 protective laws and regulations and therefore free for general use. Printed on acid-free paper The use of general descriptive names, registered names, trademarks, etc. in this publication does not or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, This work is subject to copyright. All rights are reserved, whether the whole or part of the material is Cover design : SPi Publishing Services Dr. Stefan Rostek University of Tübingen Wirtschaftswissenschaftliches Seminar Mohlstraße 36 Germany 72074 Tübingen ISBN 978-3-642-00330-1 e-ISBN 978-3-642-00331-8 DOI 1 0.1007/978-3-642-00331-8 © Springer-Verlag Berlin Heidelberg 2009 Lehrstuhl für Betriebliche Finanzwirtschaft Library of Congress Control Numbe r: "PCN applied for" ISSN 0075-8442 Springer Dordrecht Heidelberg London New York ions are liable for prosecution under the German Copyright Law. 1965, in its current version, and permissions for use must always be obtained from Springer. Violat- imply, even in the absence of a specific statement, that such names are exempt from the relevant springer is part & Springer Science+Business Media (www.springer.com) [email protected]
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To Ulrike
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Foreword Mandelbrot and van Ness (1968) suggested fractional Brownian motion as a parsimonious model for the dynamics of financial price data, which allows for dependence between returns over time. Starting with Rogers (1997) there is an ongoing dispute on the proper usage of fractional Brownian motion in option pricing theory. Problems arise because fractional Brownian motion is not a semimartingale and therefore “no arbitrage pricing” cannot be applied. While this is consensus, the consequences are not as clear. The orthodox interpretation is simply that fractional Brownian motion is an inadequate candidate for a price process. However, as shown by Cheridito (2003) any theoretical arbitrage opportunities disappear by assuming that market par- ticipants cannot react instantaneously. This is the point of departure of Rostek’s dissertation. He contributes to this research in several respects: (i) He delivers a thorough introduction to frac- tional integration calculus and uses the binomial approximation of fractional Brownian motion to give the reader a first idea of this special market setting.
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