1、n Undergraduate Introduction to Mathematics J Robert Buchanan An Undergraduate f Introduction to Financial! Mathematics This page is intentionally left blankM!MM4 J Robert Buchanan MiNersviile University, USA World Scientific NEW JERSEY LONDON - SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI Pu
2、blished by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is av
3、ailable from the British Library. AN UNDERGRADUATE INTRODUCTION TO FINANCIAL MATHEMATICS Copyright 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying,
4、recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In thi
5、s case permission to photocopy is not required from the publisher. ISBN 981-256-637-6 Printed in Singapore by World Scientific Printers (S) Pte Ltd Dedication For my wife, Monika. This page is intentionally left blankPreface This book is intended for an audience with an undergraduate level of ex pos
6、ure to calculus through elementary multivariable calculus. The book assumes no background on the part of the reader in probability or statis tics. One of my objectives in writing this book was to create a readable, reasonably self-contained introduction to financial mathematics for people wanting to
7、 learn some of the basics of option pricing and hedging. My de sire to write such a book grew out of the need to find an accessible book for undergraduate mathematics majors on the topic of financial mathematics. I have taught such a course now three times and this book grew out of my lecture notes
8、and reading for the course. New titles in financial mathemat ics appear constantly, so in the time it took me to compose this book there may have appeared several superior works on the subject. Knowing the amount of work required to produce this book, I stand in awe of authors such as those. This bo
9、ok consists of ten chapters which are intended to be read in or der, though the well-prepared reader may be able to skip the first several with no loss of understanding in what comes later. The first chapter is on interest and its role in finance. Both discretely compounded and contin uously compoun
10、ded interest are treated there. The book begins with the theory of interest because this topic is unlikely to scare off any reader no matter how long it has been since they have done any formal mathematics. The second and third chapters provide an introduction to the concepts of probability and stat
11、istics which will be used throughout the remain der of the book. Chapter Two deals with discrete random variables and emphasizes the use of the binomial random variable. Chapter Three in troduces continuous random variables and emphasizes the similarities and differences between discrete and continu
12、ous random variables. The nor-vii viii An Undergraduate Introduction to Financial Mathematics mal random variable and the close related lognormal random variable are introduced and explored in the latter chapter. In the fourth chapter the concept of arbitrage is introduced. For read ers already well
13、 versed in calculus, probability, and statistics, this is the first material which may be unfamiliar to them. The assumption that fi nancial calculations are carried out in an “arbitrage free“ setting pervades the remainder of the book. The lack of arbitrage opportunities in financial transactions e
14、nsures that it is not possible to make a risk free profit. This chapter includes a discussion of the result from linear algebra and opera tions research known as the Duality Theorem of Linear Programming. The fifth chapter introduces the reader to the concepts of random walks and Brownian motion. Th
15、e random walk underlies the mathematical model of the value of securities such as stocks and other financial instruments whose values are derived from securities. The choice of material to present and the method of presentation is difficult in this chapter due to the com plexities and subtleties of
16、stochastic processes. I have attempted to intro duce stochastic processes in an intuitive manner and by connecting elemen tary stochastic models of some processes to their corresponding determinis tic counterparts. Itos Lemma is introduced and an elementary proof of this result is given based on the
17、 multivariable form of Taylors Theorem. Read ers whose interest is piqued by material in Chapter Five should consult the bibliography for references to more comprehensive and detailed discussions of stochastic calculus. Chapter Six introduces the topic of options. Both European and Ameri can style o
18、ptions are discussed though the emphasis is on European options. Properties of options such as the Put/Call Parity formula are presented and justified. In this chapter we also derive the partial differential equation and boundary conditions used to price European call and put options. This derivatio
19、n makes use of the earlier material on arbitrage, stochastic pro cesses and the Put/Call Parity formula. The seventh chapter develops the solution to the Black-Scholes PDE. There are several different methods commonly used to derive the solution to the PDE and students benefit from different aspects
20、 of each derivation. The method I choose to solve the PDE involves the use of the Fourier Transform. Thus this chapter begins with a brief discussion of the Fourier and Inverse Fourier Transforms and their properties. Most three- or four-semester elementary calculus courses include at least an optio
21、nal section on the Fourier Transform, thus students will have the calculus background necessary to follow this discussion. It also provides exposure to the Fourier Preface IX Transform for students who will be later taking a course in PDEs and more importantly exposure for students who will not take
22、 such a course. After completing this derivation of the Black-Scholes option pricing formula students should also seek out other derivations in the literature for the purposes of comparison. Chapter Eight introduces some of the commonly discussed partial derivatives of the Black-Scholes option prici
23、ng formula. These partial derivatives help the reader to understand the sensitivity of option prices to movements in the underlying securitys value, the risk-free interest rate, and the volatility of the underlying securitys value. The collection of par tial derivatives introduced in this chapter is
24、 commonly referred to as “the Greeks“ by many financial practitioners. The Greeks are used in the ninth chapter on hedging strategies for portfolios. Hedging strategies are used to protect the value of a portfolio against movements in the underlying secu ritys value, the risk-free interest rate, and
25、 the volatility of the underlying securitys value. Mathematically the hedging strategies remove some of the low order terms from the Black-Scholes option pricing formula making it less sensitive to changes in the variables upon which it depends. Chapter Nine will discuss and illustrate several examp
26、les of hedging strategies. Chapter Ten extends the ideas introduced in Chapter Nine by model ing the effects of correlated movements in the values of investments. The tenth chapter discusses several different notions of optimality in selecting portfolios of investments. Some of the classical models
27、of portfolio selection are introduced in this chapter including the Capital Assets Pricing Model (CAPM) and the Minimum Variance Portfolio. It is the authors hope that students will find this book a useful intro duction to financial mathematics and a springboard to further study in this area. Writin
28、g this book has been hard, but intellectually rewarding work. During the summer of 2005 a draft version of this manuscript was used by the author to teach a course in financial mathematics. The author is indebted to the students of that class for finding numerous typographical errors in that earlier
29、 version which were corrected before the camera ready copy was sent to the publisher. The author wishes to thank Jill Bachstadt, Jason Buck, Mark Elicker, Kelly Flynn, Jennifer Gomulka, Nicole Hundley, Alicia Kasif, Stephen Kluth, Patrick McDevitt, Jessica Paxton, Christopher Rachor, Timothy Ren, Pa
30、mela Wentz, Joshua Wise, and Michael Zrncic. A list of errata and other information related to this book can be found at a web site I created: x An Undergraduate Introduction to Financial Mathematics http:/banach.millersville.edu/bob/book/ Please feel free to share your comments, criticism, and (I h
31、ope) praise for this work through the email address that can be found at that site. J. Robert Buchanan Lancaster, PA, USA October 31, 2005 Contents Preface vii 1. The Theory of Interest 1 1.1 Simple Interest1.2 Compound Interest 3 1.3 Continuously Compounded Interest 4 1.4 Present Value 5 1.5 Rate o
32、f Return 11 1.6 Exercises 2 2. Discrete Probability 15 2.1 Events and Probabilities 12.2 Addition Rule 7 2.3 Conditional Probability and Multiplication Rule 18 2.4 Random Variables and Probability Distributions 21 2.5 Binomial Random Variables 23 2.6 Expected Value 24 2.7 Variance and Standard Devia
33、tion 29 2.8 Exercises 32 3. Normal Random Variables and Probability 35 3.1 Continuous Random Variables 33.2 Expected Value of Continuous Random Variables 38 3.3 Variance and Standard Deviation 40 3.4 Normal Random Variables 42 3.5 Central Limit Theorem 9 xi Xll An Undergraduate Introduction to Finan
34、cial Mathematics 3.6 Lognormal Random Variables 51 3.7 Properties of Expected Value 5 3.8 Properties of Variance 58 3.9 Exercises 64. The Arbitrage Theorem 63 4.1 The Concept of Arbitrage 64.2 Duality Theorem of Linear Programming 64 4.2.1 Dual Problems 6 4.3 The Fundamental Theorem of Finance 72 4.
35、4 Exercises 75. Random Walks and Brownian Motion 77 5.1 Intuitive Idea of a Random Walk5.2 First Step Analysis 78 5.3 Intuitive Idea of a Stochastic Process 91 5.4 Stock Market Example 95 5.5 More About Stochastic Processes 97 5.6 Itos Lemma 98 5.7 Exercises 101 6. Options 103 6.1 Properties of Opti
36、ons 104 6.2 Pricing an Option Using a Binary Model 107 6.3 Black-Scholes Partial Differential Equation 110 6.4 Boundary and Initial Conditions 112 6.5 Exercises 114 7. Solution of the Black-Scholes Equation 115 7.1 Fourier Transforms 117.2 Inverse Fourier Transforms 118 7.3 Changing Variables in the
37、 Black-Scholes PDE 119 7.4 Solving the Black-Scholes Equation 122 7.5 Exercises 127 8. Derivatives of Black-Scholes Option Prices 131 8.1 Theta 138.2 Delta 3 Contents xiii 8.3 Gamma 135 8.4 Vega 6 8.5 Rho 8 8.6 Relationships Between A, 9, and T 139 8.7 Exercises 141 9. Hedging 3 9.1 General Principl
38、es 149.2 Delta Hedging 5 9.3 Delta Neutral Portfolios 149 9.4 Gamma Neutral Portfolios 151 9.5 Exercises 153 10. Optimizing Portfolios 155 10.1 Covariance and Correlation 1510.2 Optimal Portfolios 164 10.3 Utility Functions 5 10.4 Expected Utility 171 10.5 Portfolio Selection 3 10.6 Minimum Variance
39、 Analysis 177 10.7 Mean Variance Analysis 186 10.8 Exercises 191 Appendix A Sample Stock Market Data 195 Appendix B Solutions to Chapter Exercises 203 B.l The Theory of Interest 20B.2 Discrete Probability 6 B.3 Normal Random Variables and Probability 212 B.4 The Arbitrage Theorem 225 B.5 Random Walk
40、s and Brownian Motion 231 B.6 Options 23B.7 Solution of the Black-Scholes Equation 239 B.8 Derivatives of Black-Scholes Option Prices 245 B.9 Hedging 24B.10 Optimizing Portfolios 25Bibliography 265 Index 7 Chapter 1 The Theory of Interest One of the first types of investments that people learn about
41、 is some vari ation on the savings account. In exchange for the temporary use of an investors money, a bank or other financial institution agrees to pay in terest, a percentage of the amount invested, to the investor. There are many different schemes for paying interest. In this chapter we will desc
42、ribe some of the most common types of interest and contrast their differences. Along the way the reader will have the opportunity to renew their acquain tanceship with exponential functions and the geometric series. Since an amount of capital can be invested and earn interest and thus numerically in
43、crease in value in the future, the concept of present value will be in troduced. Present value provides a way of comparing values of investments made at different times in the past, present, and future. As an application of present value, several examples of saving for retirement and calculation of
44、mortgages will be presented. Sometimes investments pay the investor varying amounts of money which change over time. The concept of rate of return can be used to convert these payments in effective interest rates, making comparison of investments easier. 1.1 Simple Interest In exchange for the use o
45、f a depositors money, banks pay a fraction of the account balance back to the depositor. This fractional payment is known as interest. The money a bank uses to pay interest is generated by investments and loans that the bank makes with the depositors money. Interest is paid in many cases at specifie
46、d times of the year, but nearly always the fraction of the deposited amount used to calculate the interest is called the interest rate and is expressed as a percentage paid per year. 1 2 An Undergraduate Introduction to Financial Mathematics For example a credit union may pay 6% annually on savings
47、accounts. This means that if a savings account contains $100 now, then exactly one year from now the bank will pay the depositor $6 (which is 6% of $100) provided the depositor maintains an account balance of $100 for the entire year. In this chapter and those that follow, interest rates will be den
48、oted symbolically by r. To simplify the formulas and mathematical calculations, when r is used it will be converted to decimal form even though it may still be referred to as a percentage. The 6% annual interest rate mentioned above would be treated mathematically as r 0.06 per year. The initially d
49、eposited amount which earns the interest will be called the principal amount and will be denoted P. The sum of the principal amount and any earned interest will be called the compound amount and A will represent it symbolically. Therefore the relationship between P, r, and A for a single year period is A = P + Pr = P(l + r). The interest, once paid to the depositor, is the depositors to keep. Banks and other financial institutions “pay“ the depositor by adding the interest to the depositors account. Unless the depositor withdraws the interest or some part
