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 ofDundee K.Erdmann Oxford University L.C.G.Rogers University ofCambridge E.Süli Oxford University J.F.Toland University ofBath Other books in this series A First Course in Discrete Mathematics I.Anderson Analytic Methods for Partial Differential Equations G.Evans,J.Blackledge,P.Yardley Applied Geometry for Computer Graphics and CAD D.Marsh Basic Linear Algebra,Second Edition T.S.Blyth and E.F.Robertson Basic Stochastic Processes Z.Brzez´niak and T.Zastawniak Elementary Differential Geometry A.Pressley Elementary Number Theory G.A.Jones and J.M.Jones Elements ofAbstract Analysis M.Ó Searcóid Elements ofLogic via Numbers and Sets D.L.Johnson Essential Mathematical Biology N.F.Britton Fields,Flows and Waves:An Introduction to Continuum Models D.F.Parker Further Linear Algebra T.S.Blyth and E.F.Robertson Geometry R.Fenn Groups,Rings and Fields D.A.R.Wallace Hyperbolic Geometry J.W.Anderson Information and Coding Theory G.A.Jones and J.M.Jones Introduction to Laplace Transforms and Fourier Series P.P.G.Dyke Introduction to Ring Theory P.M.Cohn Introductory Mathematics:Algebra and Analysis G.Smith Linear Functional Analysis B.P.Rynne and M.A.Youngson Matrix Groups:An Introduction to Lie Group Theory A.Baker Measure,Integral and Probability M.Capin´ski and E.Kopp Multivariate Calculus and Geometry S.Dineen Numerical Methods for Partial Differential Equations G.Evans,J.Blackledge,P.Yardley Probability Models J.Haigh Real Analysis J.M.Howie Sets,Logic and Categories P.Cameron Special Relativity N.M.J.Woodhouse Symmetries D.L.Johnson Topics in Group Theory G.Smith and O.Tabachnikova Topologies and Uniformities I.M.James Vector Calculus P.C.Matthews Marek Capin´ski and Tomasz Zastawniak Mathematics for Finance An Introduction to Financial Engineering With 75 Figures 1 Springer Marek Capin´ski (cid:1) (cid:1) Nowy Sacz School ofBusiness–National Louis University,33-300 Nowy Sacz, ul.Zielona 27,Poland Tomasz Zastawniak Department ofMathematics,University ofHull,Cottingham Road, Kingston upon Hull,HU6 7RX,UK Cover illustration elements reproduced by kind permission of: Aptech Systems,Inc.,Publishers ofthe 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:[email protected] URL:www.aptech.com. American Statistical Association:Chance Vol 8 No 1,1995 article by KS and KW Heiner ‘Tree Rings ofthe 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 ofMathematical 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 ofa 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 ofthe 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 oftheMadelung 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,1959- 332’.0151 ISBN 1852333308 Library ofCongress Cataloging-in-Publication Data Capin´ski,Marek,1951- Mathematics for finance : an introduction to financial engineering / Marek Capin´skiand 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 ofresearch 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 ofthe publishers,or in the case ofreprographic reproduction in accordance with the terms oflicences 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 ofBertelsmannSpringer Science+Business Media GmbH http://www.springer.co.uk © Springer-Verlag London Limited 2003 Printed in the United States ofAmerica The use ofregistered names,trademarks etc.in this publication does not imply,even in the absence ofa 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 ofthe 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 Truetoitstitle,thisbookitselfisanexcellentfinancialinvestment.Fortheprice 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? Buildingonmathematicalmodelsofbondandstockprices,thesetwotheo- ries lead in different directions: Black–Scholes arbitrage pricing of options and other derivative securities on the one hand, and Markowitz portfolio optimisa- tion 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 havinganenormousimpactonthewaymodernfinancialmarketsoperate.This textbookpresentsthematalevelaimedatsecondorthirdyearundergraduate students,notonlyofmathematicsbutalso,forexample,businessmanagement, 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 ex- amples and exercises, complete with solutions, providing ample material for tutorials as well as making the book ideal for self-study. Prerequisitesincludeelementarycalculus,probabilityandsomelinearalge- bra. 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 bi- nomial and normal distributions, expectation, variance and covariance, condi- tional probability and independence. Familiarity with the Central Limit The- orem would be a bonus. In linear algebra the reader should be able to solve v vi 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 com- puter with a spreadsheet application, though this is not absolutely essential. MicrosoftExcelfileswithsolutionstoselectedexamplesandexercisesareavail- able 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 Remark4.1.Itisalsoagreatpleasuretothankourstudentsandcolleaguesfor their feedback on preliminary versions of various chapters. Readers of this book are cordially invited to visit the web page below to checkfor thelatestdownloads and corrections,ortocontacttheauthors.Your comments will be greatly appreciated. Marek Capin´ski and Tomasz Zastawniak January 2003 www.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 vii viii Contents 3.2.1 Risk-Neutral Probability ............................ 58 3.2.2 Martingale Property ................................ 61 3.3 Other Models ............................................ 63 3.3.1 Trinomial Tree Model............................... 64 3.3.2 Continuous-Time Limit ............................. 66 4. Discrete Time Market Models .............................. 73 4.1 Stock and Money Market Models........................... 73 4.1.1 Investment Strategies ............................... 75 4.1.2 The Principle of No Arbitrage ....................... 79 4.1.3 Application to the Binomial Tree Model............... 81 4.1.4 Fundamental Theorem of Asset Pricing ............... 83 4.2 Extended Models......................................... 85 5. Portfolio Management ...................................... 91 5.1 Risk .................................................... 91 5.2 Two Securities ........................................... 94 5.2.1 Risk and Expected Return on a Portfolio.............. 97 5.3 Several Securities.........................................107 5.3.1 Risk and Expected Return on a Portfolio..............107 5.3.2 Efficient Frontier ...................................114 5.4 Capital Asset Pricing Model ...............................118 5.4.1 Capital Market Line ................................118 5.4.2 Beta Factor........................................120 5.4.3 Security Market Line ...............................122 6. Forward and Futures Contracts.............................125 6.1 Forward Contracts........................................125 6.1.1 Forward Price......................................126 6.1.2 Value of a Forward Contract.........................132 6.2 Futures .................................................134 6.2.1 Pricing............................................136 6.2.2 Hedging with Futures ...............................138 7. Options: General Properties ................................147 7.1 Definitions...............................................147 7.2 Put-Call Parity ..........................................150 7.3 Bounds on Option Prices ..................................154 7.3.1 European Options ..................................155 7.3.2 European and American Calls on Non-Dividend Paying Stock .............................................157 7.3.3 American Options ..................................158 Contents ix 7.4 Variables Determining Option Prices........................159 7.4.1 European Options ..................................160 7.4.2 American Options ..................................165 7.5 Time Value of Options ....................................169 8. Option Pricing..............................................173 8.1 European Options in the Binomial Tree Model ...............174 8.1.1 One Step..........................................174 8.1.2 Two Steps.........................................176 8.1.3 General N-Step Model ..............................178 8.1.4 Cox–Ross–Rubinstein Formula .......................180 8.2 American Options in the Binomial Tree Model ...............181 8.3 Black–Scholes Formula ....................................185 9. Financial Engineering.......................................191 9.1 Hedging Option Positions..................................192 9.1.1 Delta Hedging .....................................192 9.1.2 Greek Parameters ..................................197 9.1.3 Applications .......................................199 9.2 Hedging Business Risk ....................................201 9.2.1 Value at Risk ......................................202 9.2.2 Case Study ........................................203 9.3 Speculating with Derivatives ...............................208 9.3.1 Tools .............................................208 9.3.2 Case Study ........................................209 10. Variable Interest Rates .....................................215 10.1 Maturity-Independent Yields...............................216 10.1.1 Investment in Single Bonds ..........................217 10.1.2 Duration ..........................................222 10.1.3 Portfolios of Bonds .................................224 10.1.4 Dynamic Hedging ..................................226 10.2 General Term Structure ...................................229 10.2.1 Forward Rates .....................................231 10.2.2 Money Market Account .............................235 11. Stochastic Interest Rates ...................................237 11.1 Binomial Tree Model......................................238 11.2 Arbitrage Pricing of Bonds ................................245 11.2.1 Risk-Neutral Probabilities ...........................249 11.3 Interest Rate Derivative Securities ..........................253 11.3.1 Options ...........................................254