Mathematics for Finance: An Introduction to Financial ...

[Pages:321]Mathematics for Finance: An Introduction to

Financial Engineering

Marek Capinski Tomasz Zastawniak

Springer

Springer Undergraduate Mathematics Series

Springer

London Berlin Heidelberg New York Hong Kong Milan Paris Tokyo

Advisory Board

P.J. Cameron Queen Mary and Westfield College M.A.J. Chaplain University of Dundee K. Erdmann Oxford University L.C.G. Rogers University of Cambridge E. S?li Oxford University J.F. Toland University of Bath

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Marek Capin? ski and Tomasz Zastawniak

Mathematics for Finance

An Introduction to Financial Engineering

With 75 Figures

1 Springer

Marek Capin? ski Nowy Sacz School of Business?National Louis University, 33-300 Nowy Sacz, ul. Zielona 27, Poland Tomasz Zastawniak Department of Mathematics, University of Hull, Cottingham Road, Kingston upon Hull, HU6 7RX, UK

Cover illustration elements reproduced by kind permission of: Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E. Kent-Kangley Road, Maple Valley, WA 98038,

USA. Tel: (206) 432 - 7855 Fax (206) 432 - 7832 email: info@ URL: . American Statistical Association: Chance Vol 8 No 1, 1995 article by KS and KW Heiner `Tree Rings of the Northern Shawangunks' page 32

fig 2. Springer-Verlag: Mathematica in Education and Research Vol 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor

`Illustrated Mathematics: Visualization of Mathematical Objects' page 9 fig 11, originally published as a CD ROM `Illustrated Mathematics' by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4. Mathematica in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate `Traffic Engineering with Cellular Automata' page 35 fig 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott `The Implicitization of a Trefoil Knot' page 14. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola `Coins, Trees, Bars and Bells: Simulation of the Binomial Process' page 19 fig 3. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Richard Gaylord and Kazume Nishidate `Contagious Spreading' page 33 fig 1. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Joe Buhler and Stan Wagon `Secrets of theMadelung Constant' page 50 fig 1.

British Library Cataloguing in Publication Data Capin?ski, Marek, 1951-

Mathematics for finance : an introduction to financial engineering. - (Springer undergraduate mathematics series) 1. Business mathematics 2. Finance ? Mathematical models I. Title II. Zastawniak, Tomasz, 1959332'.0151 ISBN 1852333308

Library of Congress Cataloging-in-Publication Data

Capin?ski, Marek, 1951-

Mathematics for finance : an introduction to financial engineering / Marek Capin?ski and

Tomasz Zastawniak.

p. cm. -- (Springer undergraduate mathematics series)

Includes bibliographical references and index.

ISBN 1-85233-330-8 (alk. paper)

1. Finance ? Mathematical models. 2. Investments ? Mathematics. 3. Business

mathematics. I. Zastawniak, Tomasz, 1959- II. Title. III. Series.

HG106.C36 2003

332.6'01'51--dc21

2003045431

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 1-85233-330-8 Springer-Verlag London Berlin Heidelberg a member of BertelsmannSpringer Science+Business Media GmbH

? Springer-Verlag London Limited 2003 Printed in the United States of America

The use of 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 laws and regulations and therefore free for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Typesetting: Camera ready by the authors 12/3830-543210 Printed on acid-free paper SPIN 10769004

Preface

True to its title, this book itself is an excellent financial investment. For the price of one volume it teaches two Nobel Prize winning theories, with plenty more included for good measure. How many undergraduate mathematics textbooks can boast such a claim?

Building on mathematical models of bond and stock prices, these two theories lead in different directions: Black?Scholes arbitrage pricing of options and other derivative securities on the one hand, and Markowitz portfolio optimisation and the Capital Asset Pricing Model on the other hand. Models based on the principle of no arbitrage can also be developed to study interest rates and their term structure. These are three major areas of mathematical finance, all having an enormous impact on the way modern financial markets operate. This textbook presents them at a level aimed at second or third year undergraduate students, not only of mathematics but also, for example, business management, finance or economics.

The contents can be covered in a one-year course of about 100 class hours. Smaller courses on selected topics can readily be designed by choosing the appropriate chapters. The text is interspersed with a multitude of worked examples and exercises, complete with solutions, providing ample material for tutorials as well as making the book ideal for self-study.

Prerequisites include elementary calculus, probability and some linear algebra. In calculus we assume experience with derivatives and partial derivatives, finding maxima or minima of differentiable functions of one or more variables, Lagrange multipliers, the Taylor formula and integrals. Topics in probability include random variables and probability distributions, in particular the binomial and normal distributions, expectation, variance and covariance, conditional probability and independence. Familiarity with the Central Limit Theorem would be a bonus. In linear algebra the reader should be able to solve

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Mathematics for Finance

systems of linear equations, add, multiply, transpose and invert matrices, and compute determinants. In particular, as a reference in probability theory we recommend our book: M. Capin?ski and T. Zastawniak, Probability Through Problems, Springer-Verlag, New York, 2001.

In many numerical examples and exercises it may be helpful to use a computer with a spreadsheet application, though this is not absolutely essential. Microsoft Excel files with solutions to selected examples and exercises are available on our web page at the addresses below.

We are indebted to Nigel Cutland for prompting us to steer clear of an inaccuracy frequently encountered in other texts, of which more will be said in Remark 4.1. It is also a great pleasure to thank our students and colleagues for their feedback on preliminary versions of various chapters.

Readers of this book are cordially invited to visit the web page below to check for the latest downloads and corrections, or to contact the authors. Your comments will be greatly appreciated.

Marek Capin?ski and Tomasz Zastawniak January 2003

springer.co.uk/M4F

Contents

1. Introduction: A Simple Market Model . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Basic Notions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 No-Arbitrage Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 One-Step Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Risk and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Call and Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Managing Risk with Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2. Risk-Free Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.2 Periodic Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 Streams of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.4 Continuous Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.5 How to Compare Compounding Methods . . . . . . . . . . . . . . 35 2.2 Money Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.2 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.3 Money Market Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3. Risky Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Dynamics of Stock Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.1 Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.2 Expected Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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