Mathematics for Finance

321 Pages·2006·6.5 MB·English

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
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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
(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:
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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
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http://www.springer.co.uk
© Springer-Verlag London Limited 2003
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The use ofregistered names,trademarks etc.in this publication does not imply,even in the absence ofa specific
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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