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Springer Finance
Editorial Board
M. Avellaneda
G. Barone-Adesi
M. Broadie
M.H.A. Davis
E. Derman
C. Kl¨ppelberg
E. Kopp
W. Schachermayer
Robert J. Elliott and P. Ekkehard Kopp
Mathematics of
Financial Markets
Second edition
 Robert J. Elliott P. Ekkehard Kopp
Haskayne School of Business Department of Mathematics
University of Calgary University of Hull
Calgary, Alberta Hull HU6 7RX
Canada T2N 1N4 Yorkshire
robert.elliott@haskayne.ucalgary.ca United Kingdom
p.e.kopp@hull.ac.uk
With 7 figures.
Library of Congress Cataloging-in-Publication Data
Elliott, Robert J. (Robert James), 1940–
Mathematics of financial markets / Robert J. Elliott and P. Ekkehard Kopp.—2nd ed.
p. cm. — (Springer finance)
Includes bibliographical references and index.
ISBN 0-387-21292-2
1. Investments—Mathematics. 2. Stochastic analysis. 3. Options
(Finance)—Mathematical models. 4. Securities—Prices—Mathematical models.
I. Kopp, P. E., 1944– II. Title. III. Series.
HG4515.3.E37 2004
332.6′01′51—dc22
2004052557
ISBN 0-387-21292-2
Printed on acid-free paper.
© 2005 Springer Science+Business Media Inc.
All rights reserved. This work may not be translated or copied in whole or in part without
the written permission of the publisher (Springer Science+Business Media Inc., 233
Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with
reviews or scholarly analysis. Use in connection with any form of information storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar meth-
odology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms,
even if they are not identified as such, is not to be taken as an expression of opinion as
to whether or not they are subject to proprietary rights.
Printed in the United States of America.
(EB)
987654321 SPIN 10936511
springeronline.com
Preface
This work is aimed at an audience with a sound mathematical background
wishing to learn about the rapidly expanding field of mathematical finance.
Its content is suitable particularly for graduate students in mathematics
who have a background in measure theory and probability.
The emphasis throughout is on developing the mathematical concepts
required for the theory within the context of their application. No attempt
is made to cover the bewildering variety of novel (or ‘exotic’) financial in-
struments that now appear on the derivatives markets; the focus through-
out remains on a rigorous development of the more basic options that lie
at the heart of the remarkable range of current applications of martingale
theory to financial markets.
The first five chapters present the theory in a discrete-time framework.
Stochastic calculus is not required, and this material should be accessible
to anyone familiar with elementary probability theory and linear algebra.
The basic idea of pricing by arbitrage (or, rather, by non-arbitrage)
is presented in Chapter 1. The unique price for a European option in a
single-period binomial model is given and then extended to multi-period
binomial models. Chapter 2 introduces the idea of a martingale measure
for price processes. Following a discussion of the use of self-financing trad-
ing strategies to hedge against trading risk, it is shown how options can
be priced using an equivalent measure for which the discounted price pro-
cess is a martingale. This is illustrated for the simple binomial Cox-Ross-
Rubinstein pricing models, and the Black-Scholes formula is derived as the
limit of the prices obtained for such models. Chapter 3 gives the ‘funda-
mental theorem of asset pricing’, which states that if the market does not
contain arbitrage opportunities there is an equivalent martingale measure.
Explicit constructions of such measures are given in the setting of finite
market models. Completeness of markets is investigated in Chapter 4; in a
complete market, every contingent claim can be generated by an admissible
self-financing strategy (and the martingale measure is unique). Stopping
times, martingale convergence results, and American options are discussed
in a discrete-time framework in Chapter 5.
The second five chapters of the book give the theory in continuous time.
This begins in Chapter 6 with a review of the stochastic calculus. Stopping
times, Brownian motion, stochastic integrals, and the Ito differentiation
v
vi
Preface
rule are all defined and discussed, and properties of stochastic differential
equations developed.
The continuous-time pricing of European options is developed in Chap-
ter 7. Girsanov’s theorem and martingale representation results are de-
veloped, and the Black-Scholes formula derived. Optimal stopping results
are applied in Chapter 8 to a thorough study of the pricing of American
options, particularly the American put option.
Chapter 9 considers selected results on term structure models, forward
and future prices, and change of numeraire, while Chapter 10 presents the
basic framework for the study of investment and consumption problems.
Acknowledgments
Sections of the book have been presented in courses
at the Universities of Adelaide and Alberta. The text has consequently
benefited from subsequent comments and criticism. Our particular thanks
go to Monique Jeanblanc-Pique, whose careful reading of the text and
valuable comments led to many improvements. Many thanks are also due
to Volker Wellmann for reading much of the text and for his patient work
in producing consistent T
E
X files and the illustrations.
Finally, the authors wish to express their sincere thanks to the Social
Sciences and Humanities Research Council of Canada for its financial sup-
port of this project.
Edmonton, Alberta, Canada
Robert J. Elliott
Hull, United Kingdom
P. Ekkehard Kopp
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