Aspects of Mathematical Finance
Considering the stupendous gain in importance, in the banking and insurance industries since the early 1990’s, of mathematical methodology, especially probabilistic methodology, it was a very natural idea for the French "Académie des Sciences" to propose
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Aspects of Mathematical Finance
Aspects of Mathematical Finance
Marc Yor, Editor
Aspects of Mathematical Finance
Marc Yor Laboratoire de Probabilités et Modèles Aléatoires Université Pierre et Marie Curie Boîte courrier 188 75252 Paris Cedex 05 France
Chapters 1, 2, 6, 7 translated from the French original by Kathleen Qechar. Original French edition “Aspects des mathématiques financières” published by Lavoisier, Paris 2006.
ISBN: 978-3-540-75258-5
e-ISBN: 978-3-540-75265-3
Library of Congress Control Number: 2007941932 Mathematics Subject Classification (2000): 90C15, 90C48, 91B30, 90C46 c 2008 Springer-Verlag Berlin Heidelberg ° 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: WMX Design GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Contents
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction: Some Aspects of Financial Mathematics . . . . . . . . . . . . . . . . M. Yor
1
Financial Uncertainty, Risk Measures and Robust Preferences . . . . . . . . . . H. F¨ollmer
3
The Notion of Arbitrage and Free Lunch in Mathematical Finance . . . . . . 15 W. Schachermayer Dynamic Financial Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 P. Barrieu and N. El Karoui Stochastic Clock and Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 H. Geman Options and Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 53 D. Lamberton Mathematics and Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 E. Gobet, G. Pag`es, and M. Yor Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
Contributors
Pauline Barrieu Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK, [email protected] Nicole El Karoui ´ C.M.A.P., Ecole Polytechnique, 91128 Palaiseau C´edex, France, [email protected] Hans Foellmer Institut f¨ur Mathematik, Humboldt Universi
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