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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)
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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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