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Finance/Mathematics A First Course in Financial Mathematics Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives. It examines the principles of option pricing in detail via standard binomial and stochastic calculus models. Developing the requisite mathematical background as needed, the text introduces probability theory and stochastic calculus at an undergraduate level. Hugo D. Junghenn Option Valuation A First Course in Financial Mathematics Junghenn Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective. Numerous examples and exercises help readers gain expertise with financial calculus methods and increase their general mathematical sophistication. The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments. Hints and solutions to odd-numbered problems are given in an appendix. A First Course in Financial Mathematics The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model. The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools. The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs. The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black– Scholes–Merton model. Option Valuation Option Valuation K14090 K14090_Cover.indd 1 10/7/11 11:23 AM Option Valuation A First Course in Financial Mathematics CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector. This series aims to capture new developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and concrete realworld examples is highly encouraged. Series Editors M.A.H. Dempster Dilip B. Madan Rama Cont Centre for Financial Research Department of Pure Mathematics and Statistics University of Cambridge Robert H. Smith School of Business University of Maryland Center for Financial Engineering Columbia University New York Published Titles American-Style Derivatives; Valuation and Computation, Jerome Detemple Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,  Pierre Henry-Labordère Credit Risk: Models, Derivatives, and Management, Niklas Wagner Engineering BGM, Alan Brace Financial Modelling with Jump Processes, Rama Cont and Peter Tankov Interest Rate Modeling: Theory and Practice, Lixin Wu Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and  Christoph Wagner Introduction to Stochastic Calculus Applied to Finance, Second Edition,  Damien Lamberton and Bernard Lapeyre Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,  and Gerald Kroisandt Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn Portfolio Optimization and Performance Analysis, Jean-Luc Prigent Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers Stochastic Finance: A Numeraire Approach, Jan Vecer Stochastic Financial Models, Douglas Kennedy Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy Unravelling the Credit Crunch, David Murphy Proposals for the series should be submitted to one of the series editors above or directly to: CRC Press, Taylor & Francis Group 4th, Floor, Albert House 1-4 Singer Street London EC2A 4BQ UK Option Valuation A First Course in Financial Mathematics Hugo D. Junghenn CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150312 International Standard Book Number-13: 978-1-4398-8912-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com TO MY FAMILY Mary, Katie, Patrick, Sadie v This page intentionally left blank Contents xi Preface 1 Interest and Present Value 1.1 Compound Interest 1.2 Annuities 1.3 Bonds 1.4 Rate of Return 1.5 Exercises 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Probability Spaces 13 2.1 Sample Spaces and Events . . . . . . . . . . . . . . . . . . . 13 2.2 Discrete Probability Spaces . . . . . . . . . . . . . . . . . . . 14 2.3 General Probability Spaces . . . . . . . . . . . . . . . . . . . 16 2.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 20 2.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Exercises 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Random Variables 27 3.1 Denition and General Properties . . . . . . . . . . . . . . . 3.2 Discrete Random Variables 3.3 Continuous Random Variables . . . . . . . . . . . . . . . . . 32 3.4 Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Independent Random Variables . . . . . . . . . . . . . . . . 35 3.6 Sums of Independent Random Variables . . . . . . . . . . . . 38 3.7 Exercises 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Options and Arbitrage 27 29 43 4.1 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classication of Derivatives 44 4.3 Forwards 4.4 Currency Forwards . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.6 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Properties of Options 4.8 Dividend-Paying Stocks 4.9 Exercises . . . . . . . . . . . . . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 50 . . . . . . . . . . . . . . . . . . . . . . 53 . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 vii viii 5 Discrete-Time Portfolio Processes 59 5.1 Discrete-Time Stochastic Processes. . . . . . . . . . . . . . . 5.2 Self-Financing Portfolios 59 . . . . . . . . . . . . . . . . . . . . 5.3 Option Valuation by Portfolios 61 5.4 Exercises . . . . . . . . . . . . . . . . . 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Expectation of a Random Variable 67 6.1 Discrete Case: Denition and Examples . . . . . . . . . . . . 6.2 Continuous Case: Denition and Examples 6.3 Properties of Expectation . . . . . . . . . . . . . . . . . . . . 69 6.4 Variance of a Random Variable . . . . . . . . . . . . . . . . . 71 6.5 The Central Limit Theorem . . . . . . . . . . . . . . . . . . 73 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . 7 The Binomial Model 67 68 77 7.1 Construction of the Binomial Model . . . . . . . . . . . . . . 7.2 Pricing a Claim in the Binomial Model 7.3 The Cox-Ross-Rubinstein Formula 7.4 Exercises 77 . . . . . . . . . . . . 80 . . . . . . . . . . . . . . . 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8 Conditional Expectation and Discrete-Time Martingales 89 8.1 Denition of Conditional Expectation . . . . . . . . . . . . . 8.2 Examples of Conditional Expectation . . . . . . . . . . . . . 92 8.3 Properties of Conditional Expectation . . . . . . . . . . . . . 94 8.4 Discrete-Time Martingales . . . . . . . . . . . . . . . . . . . 96 8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 9 The Binomial Model Revisited 89 101 9.1 Martingales in the Binomial Model . . . . . . . . . . . . . . 9.2 Change of Probability 9.3 American Claims in the Binomial Model 9.4 Stopping Times 9.5 Optimal Exercise of an American Claim 9.6 . . . . . . . . . . . . . . . . . . . . . . 101 103 . . . . . . . . . . . 105 . . . . . . . . . . . . . . . . . . . . . . . . . 108 . . . . . . . . . . . . 111 Dividends in the Binomial Model . . . . . . . . . . . . . . . 114 9.7 The General Finite Market Model . . . . . . . . . . . . . . . 115 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10 Stochastic Calculus 119 10.1 Dierential Equations . . . . . . . . . . . . . . . . . . . . . . 119 10.2 Continuous-Time Stochastic Processes . . . . . . . . . . . . . 120 10.3 Brownian Motion 122 . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Variation of Brownian Paths . . . . . . . . . . . . . . . . . . 123 . . . . . . . . . . . . . . . . . . . 126 . . . . . . . . . . . . . . . . . . . . . . . 126 10.5 Riemann-Stieltjes Integrals 10.6 Stochastic Integrals 10.7 The Ito-Doeblin Formula . . . . . . . . . . . . . . . . . . . . 10.8 Stochastic Dierential Equations . . . . . . . . . . . . . . . . 131 136 ix 10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Black-Scholes-Merton Model 11.1 The Stock Price SDE 141 . . . . . . . . . . . . . . . . . . . . . . 11.2 Continuous-Time Portfolios . . . . . . . . . . . . . . . . . . . 11.3 The Black-Scholes-Merton PDE . . . . . . . . . . . . . . . . 11.4 Properties of the BSM Call Function 11.5 Exercises 139 141 142 143 . . . . . . . . . . . . . 146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 12 Continuous-Time Martingales 12.1 Conditional Expectation 151 . . . . . . . . . . . . . . . . . . . . 12.2 Martingales: Denition and Examples 152 . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . . . . 156 12.3 Martingale Representation Theorem 12.4 Moment Generating Functions 151 . . . . . . . . . . . . . 12.5 Change of Probability and Girsanov's Theorem . . . . . . . . 158 12.6 Exercises 161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The BSM Model Revisited 163 13.1 Risk-Neutral Valuation of a Derivative 13.2 Proofs of the Valuation Formulas 13.3 Valuation under P . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . . . . . . 167 13.4 The Feynman-Kac Representation Theorem 13.5 Exercises 163 . . . . . . . . . . . . . . . . . . . . . . . . 168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 14 Other Options 173 14.1 Currency Options . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Forward Start Options 14.3 Chooser Options 173 . . . . . . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . . . . . . . . . . . 176 14.4 Compound Options . . . . . . . . . . . . . . . . . . . . . . . 14.5 Path-Dependent Derivatives 177 . . . . . . . . . . . . . . . . . . 178 14.5.1 Barrier Options . . . . . . . . . . . . . . . . . . . . . . 179 14.5.2 Lookback Options . . . . . . . . . . . . . . . . . . . . 185 . . . . . . . . . . . . . . . . . . . . . . 191 14.5.3 Asian Options 14.6 Quantos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.7 Options on Dividend-Paying Stocks 197 14.7.1 Continuous Dividend Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 14.7.2 Discrete Dividend Stream . . . . . . . . . . . . . . . . 198 14.8 American Claims in the BSM Model 14.9 Exercises . . . . . . . . . . . . . . 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A Sets and Counting 209 B Solution of the BSM PDE 215 C Analytical Properties of the BSM Call Function 219 x D Hints and Solutions to Odd-Numbered Problems 225 Bibliography 247 Index 249 xi Preface This text is intended as an introduction to the mathematics and models used in the valuation of nancial derivatives. It is designed for an audience with a background in standard multivariable calculus. Otherwise, the book is essentially self-contained: The requisite probability theory is developed from rst principles and introduced as needed, and nance theory is explained in detail under the assumption that the reader has no background in the subject. The book is an outgrowth of a set of notes developed for an undergraduate course in nancial mathematics oered at The George Washington University. The course serves mainly majors in mathematics, economics, or nance and is intended to provide a straightforward account of the principles of option pricing. The primary goal of the text is to examine these principles in detail via the standard binomial and stochastic calculus models. Of course, a rigorous exposition of such models requires a coherent development of the requisite mathematical background, and it is an equally important goal to provide this background in a careful manner consistent with the scope of the text. Indeed, it is hoped that the text may serve as an introduction to applied probability (through the lens of mathematical nance). The book consists of fourteen chapters, the rst nine of which develop option valuation techniques in discrete time, the last ve describing the theory in continuous time. The emphasis is on two models, the (discrete time) binomial model and the (continuous time) Black-Scholes-Merton model. The binomial model serves two purposes: First, it provides a practical way to price options using relatively elementary mathematical tools. Second, it allows a straightforward and concrete exposition of fundamental principles of nance, such as arbitrage and hedging, without the possible distraction of complex mathematical constructs. Many of the ideas that arise in the binomial model foreshadow notions inherent in the more mathematically sophisticated BlackScholes-Merton model. Chapter 1 gives an elementary account of present value. Here the focus is on risk-free investments, such money market accounts and bonds, whose values are determined by an interest rate. Investments of this type provide a way to measure the value of a risky asset, such as a stock or commodity, and mathematical descriptions of such investments form an important component of option pricing techniques. Chapters 2, 3, and 6 form the core of the general probability portion of the text. The exposition is self-contained and uses only basic combinatorics and elementary calculus. Appendix A provides a brief overview of the elementary set theory and combinatorics used in these chapters. Readers with a good background in probability may safely give this part of the text a cursory reading. While our approach is largely standard, the more sophisticated notions of event σ -eld and ltration are introduced early to prepare the reader xii for the martingale theory developed in later chapters. We have avoided using Lebesgue integration by considering only discrete and continuous random variables. Chapter 4 describes the most common types of nancial derivatives and emphasizes the role of arbitrage in nance theory. The assumption of an arbitrage-free market, that is, one that allows no free lunch, is crucial in developing useful pricing models. An important consequence of this assumption is the put-call parity formula, which relates the cost of a standard call option to that of the corresponding put. Discrete-time stochastic processes are introduced in Chapter 5 to provide a rigorous mathematical framework for the notion of a self-nancing portfolio. The chapter describes how such portfolios may be used to replicate options in an arbitrage-free market. Chapter 7 introduces the reader to the binomial model. The main result is the construction of a replicating, self-nancing portfolio for a general European claim. The most important consequence is the Cox-Ross-Rubinstein formula for the price of a call option. Chapter 9 considers the binomial model from the vantage point of discrete-time martingale theory, which is developed in Chapter 8, and takes up the the more dicult problem of pricing and hedging an American claim. Chapter 10 gives an overview of Brownian motion, constructs the Ito integral for processes with continuous paths, and uses Ito's formula to solve various stochastic dierential equations. Our approach to stochastic calculus builds on the reader's knowledge of classical calculus and emphasizes the similarities and dierences between the two theories via the notion of variation of a function. Chapter 11 uses the tools developed in Chapter 10 to construct the BlackScholes-Merton PDE, the solution of which leads to the celebrated BlackScholes formula for the price of a call option. A detailed analysis of the analytical properties of the formula is given in the last section of the chapter. The more technical proofs are relegated to appendices so as not to interrupt the main ow of ideas. Chapter 12 gives a brief overview of those aspects of continuous-time martingales needed for risk-neutral pricing. The primary result is Girsanov's Theorem, which guarantees the existence of risk-neutral probability measures. Chapters 13 and 14 provide a martingale approach to option pricing, using risk-neutral probability measures to nd the value of a variety of derivatives, including path-dependent options. Rather than being encyclopedic, the material is intended to convey the essential ideas of derivative pricing and to demonstrate the utility and elegance of martingale techniques in this endeavor. The text contains numerous examples and 200 exercises designed to help the reader gain expertise in the methods of nancial calculus and, not incidentally, to increase his or her level of general mathematical sophistication. The exercises range from routine calculations to spreadsheet projects to the xiii pricing of a variety of complex nancial instruments. Hints and solutions to the odd-numbered problems are given in Appendix D. For greater clarity and ease of exposition (and to remain within the intended scope of the text), we have avoided stating results in their most general form. Thus, interest rates are assumed to be constant, paths of stochastic processes are required to be continuous, and nancial markets trade in a single risky asset. While these assumptions may be unrealistic, it is our belief that the reader who has obtained a solid understanding of the theory in this simplied setting will have little diculty in making the transition to more general contexts. While the text contains numerous examples and problems involving the use of spreadsheets, we have not included any discussion of general numerical techniques, as there are several excellent texts devoted to this subject. Indeed, such a text could be used to good eect in conjunction with the present one. It is inevitable that any serious development of option pricing methods at the intended level of this book must occasionally resort to invoking a result that falls outside the scope of the text. For the few times that this has occurred, we have tried either to give a sketch of the proof or, failing that, to give references, general or specic, where the reader may nd a reasonably accessible proof. The text is organized to allow as exible use as possible. The precursor to the book, in the form of a set of notes, has been successfully tested in the classroom as a single semester course in discrete-time theory only (Chapters 19) and as a one-semester course giving an overview of both discrete-time and continuous-time models (Chapters 17, 10, and 11). It may also easily serve as a two-semester course, with Chapters 113 forming the core and selections from Chapter 14. To the students whose sharp eye caught typos, inconsistencies, and downright errors in the notes leading up to the book: thank you. To the readers of this text: the author would be grateful indeed for similar observations, should the opportunity arise, as well as for suggestions for improvements. Hugo D. Junghenn Washington, D.C., USA This page intentionally left blank Chapter 1 Interest and Present Value In this chapter, we consider assets whose value is determined by an interest rate. If the asset is guaranteed, as in the case of an insured savings account or a government bond (which, typically, has only a small likelihood of default), the asset is said to be risk-free . Such an asset stands in contrast to a risky asset, for example, a stock or commodity, whose future values cannot be determined with certainty. As we shall see in later chapters, mathematical models that describe the value of a risky asset typically include a component involving a risk-free asset. Therefore, our rst goal is to describe how risk-free assets are valued. 1.1 Compound Interest Interest is a fee paid by one party for the use of cash assets of another. The amount of interest is generally time dependent: the longer the outstanding balance, the more interest is accrued. A familiar example is the interest generated by a money market account. The bank pays the depositor an amount that is usually a fraction of the balance in the account, that fraction given in terms of a prorated annual percentage called the nominal rate. n = 1, 2, . . .. Suppose the initial deposit is A0 and the interest rate per period is i. If interest is compounded , then, after the rst period, the value of the account is A1 = A0 + iA0 = A0 (1 + i), after the second period the value is A2 = A1 + iA1 = A1 (1 + i) = A0 (1 + i)2 , and so on. In general, the value of the account at time n is Consider rst an account that pays interest at the discrete times An = A0 (1 + i)n , A0 is called the future value. present value or n = 0, 1, 2, . . . . discounted value Now suppose that the nominal rate is times a year. Then i = r/m r (1.1) of the account and An and interest is compounded hence the value of the account after At = A0 (1 + r/m)mt . t a m years is (1.2) 1 Option Valuation: A First Course in Financial Mathematics 2 The distinction between the formulas (1.1) and (1.2) is that the former expresses the value of the account as a function of the number of compounding intervals (that is, at the discrete times a function of continuous time t n), while the latter gives the value as (in years). In contrast to an account earning compound interest, an account drawing simple interest has time-t value At = A0 (1 + tr). In this case, interest is calculated only on the initial deposit A0 and not on the preceding account value. Example 1.1.1. Table 1.2 gives the value after two years of an account with present value $800. The account is assumed to earn interest at an annual rate of 12%. Value 800(1.12)2 800(1.06)4 800(1.03)8 800(1.01)24 800(1.0003)730 Compound Method = $1,003.52 annually = $1,009.98 semiannually = $1,013.42 quarterly = $1,015.79 monthly = $1,016.96 daily TABLE 1.1: Account Value in Two Years Note that for simple interest the value of the account after two years is 800(1.24) = $992.00. The above example suggests that compounding more frequently results in (1+r/m)m x = m/r in (1.2) a greater return. This is can be seen from the fact that the sequence is increasing in m. To see what happens when so that m → ∞, set rt At = A0 [(1 + 1/x)x ] . As m → ∞, l'Hospital's rule shows that (1 + 1/x)x → e. In this way, we arrive at the formula for continuously compounded interest : At = A0 ert . (1.3) Returning to Example 1.1.1 we see that, if interest is compounded continuously, then the value of the account after two years is 800e(.12)2 = $1, 016.99, not signicantly more than for daily compounding. The eective interest rate re is the simple interest rate that produces the Interest and Present Value 3 same yield in one year as compound interest. If interest is compounded times a year, this means that A0 (1 + r/m)m = A0 (1 + re ) m hence re = (1 + r/m)m − 1. If interest is compounded continuously, then A0 er = A0 (1 + re ) so that re = er − 1. Example 1.1.2. You just inherited $10,000, which you decide to deposit in one of three banks, A, B, or C. Bank A pays 11% compounded semiannually, bank B pays 10.76% compounded monthly, and bank C pays 10.72 % compounded continuously. Which bank should you choose? Solution: We compute the eective rate re for each given interest rate. Rounding, we have = (1 + .11/2)2 − 1 = 0.113025 = (1 + .1076/12)12 − 1 = 0.113068 = e.1072 − 1 = 0.113156 re re re for Bank A, for Bank B, for Bank C. Bank C has the highest eective rate and is therefore the best choice. 1.2 Annuities An annuity is a sequence of periodic payments of a xed amount, say, P. The payments may take the form of deposits into an account, such as a pension fund or layaway plan, or withdrawals from an account, for example, a trust fund or retirement account. annual rate r compounded 1 Suppose that the account pays interest at an m times per year and that a deposit (withdrawal) is made at the end of each compounding interval. We seek the value account at time n, that is, immediately after the nth An of the payment. An is the sum of the time-n values of payments 1 j accrues interest over n − j payment periods, its P (1 + r/m)n−j . Thus, In the case of deposits, through n. Since payment time-n value is An = P (1 + x + x2 + · · · + xn−1 ), The geometric series sums to (xn − 1)/(x − 1), An = P 1 An (1 + i)n − 1 , i x := 1 + r . m hence i := r . m (1.4) account into which periodic deposits are made for the purpose of retiring a debt or purchasing an asset is sometimes called a sinking fund. Option Valuation: A First Course in Financial Mathematics 4 For withdrawals we argue as follows: Let account. The value at the end of period nth payment, is An−1 plus the interest withdrawal reduces that value by P n, iAn−1 A0 be the initial value of the just before withdrawal of the over that period. Making the so An = aAn−1 − P, a := 1 + i. Iterating, we obtain An = a2 An−2 − (1 + a)P = · · · = an A0 − (1 + a + a2 + · · · + an−1 )P. Thus, 1 − (1 + i)n i (1 + i)n (iA0 − P ) + P . = i An = (1 + i)n A0 + P (1.5) Now assume that the account is drawn down to zero after Setting n=N and AN = 0 in (1.5) and solving for A0 = P A0 1 − (1 + i)−N . i This is the initial deposit required to support exactly P N withdrawals. yields (1.6) N withdrawals of amount from, say, a retirement account or trust fund. It may be seen as the sum of the present values of the Solving for P N withdrawals. in (1.6) we obtain P = A0 i , 1 − (1 + i)−N (1.7) which may be used, for example, to calculate the mortgage payment for a A0 mortgage of size (see Example 1.2.2, below). Substituting (1.7) into (1.5) we obtain the following formula for the time-n value of an annuity supporting exactly N withdrawals: An = A0 Example 1.2.1. size P After for 1 − (1 + i)n−N , 1 − (1 + i)−N n = 0, 1, . . . , N. (1.8) (Retirement plan). Suppose you make monthly deposits of r, compounded monthly. t years you wish to make monthly withdrawals of size Q from the account into a retirement account with an annual rate s years, drawing down the account to zero. By (1.4) and (1.6) it must then be the case that P 1 − (1 + i)−12s (1 + i)12t − 1 =Q , i i i := r , 12 Interest and Present Value or P 1 − (1 + i)−12s = . Q (1 + i)12t − 1 For a numerical example, suppose that t = 40, s = 30, 5 (1.9) and r = .06. Then P 1 − (1.005)−360 = ≈ .084, Q (1.005)480 − 1 so that a withdrawal of, say, Q = $5000 during retirement would require monthly deposits of P = (.084)5000 ≈ $419. A more realistic analysis takes into account the reduction of purchasing power due to ination. Suppose that ination is running at 3% per year or .25% per month. This means that goods and services that cost $1 now will cost $(1.0025) n n months from now. The present value purchasing power of the rst withdrawal is then 5000(1.0025)−481 ≈ $1504, while that of the last withdrawal is only 5000(1.0025)−840 ≈ $614. For the rst withdrawal to have the current purchasing power of $5000, Q would have to be 5000(1.0025)481 ≈ $16, 617, which would require monthly deposits of P = (.084)16, 617 ≈ $1396. For the last withdrawal to have the current purchasing power of $5000, Q would have to be 5000(1.0025)840 ≈ $40, 724, requiring monthly deposits of P = (.084)40, 724 ≈ $3421, more than eight times the amount calculated without considering ination! Example 1.2.2. (Amortization). Suppose you take out a 20-year, $200,000 mortgage at an annual rate of 8% compounded monthly. Your monthly mort- P constitute an N = 240. Here An is A0 = $200, 000, i = .08/12 = n. By gage payments annuity with .0067, the amount owed at the end of month and (1.7), the mortgage payments are P = 200, 000 .0067 = $1677.85. 1 − (1.0067)−240
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