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
To k yo
Advisory Board
P.J. Cameron Queen Mary and Westfield College
M.A.J. Chaplain University of Dundee
K. Erdmann Oxford University
L.C.G. Rogers University of Cambridge
E. Süli Oxford University
J.F. Toland University of Bath
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. Brze´zniak and T. Zastawniak
Elementary Differential Geometry A. Pressley
Elementary Number Theory G.A. Jones and J.M. Jones

1 Springer
Marek Capi´nski
Nowy Sa

cz School of Business–National Louis University, 33-300 Nowy Sa

cz,
ul. Zielona 27, Poland
Tomasz Zastawniak
Department of Mathematics, University of Hull, Cottingham Road,
Kingston upon Hull, HU6 7RX, UK
Cover illustration elements reproduced by kind permission of:
Aptech Systems, Inc., Publishers of the 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: URL: www.aptech.com.
American Statistical Association: Chance Vol 8 No 1, 1995 article by KS and KW Heiner ‘Tree Rings of the 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 of Mathematical 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
of a 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 of the 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 of theMadelung Constant’ page 50 fig 1.
British Library Cataloguing in Publication Data
Capi´nski, Marek, 1951-
Mathematics for finance : an introduction to financial
engineering. - (Springer undergraduate mathematics series)

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
True to its title, this book itself is an excellent financial investment. For the price
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?
Building on mathematical models of bond and stock prices, these two theo-
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
having an enormous impact on the way modern financial markets operate. This
textbook presents them at a level aimed at second or third year undergraduate
students, not only of mathematics but also, for example, business management,
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.
Prerequisites include elementary calculus, probability and some linear alge-
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-

1.7 Managing Risk with Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Risk-Free Assets 21
2.1 TimeValueofMoney 21
2.1.1 SimpleInterest 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-CouponBonds 39
2.2.2 CouponBonds 41
2.2.3 MoneyMarketAccount 43
3. Risky Assets 47
3.1 DynamicsofStockPrices 47
3.1.1 Return 49
3.1.2 ExpectedReturn 53
3.2 BinomialTreeModel 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 TrinomialTree Model 64
3.3.2 Continuous-TimeLimit 66
4. Discrete Time Market Models 73
4.1 Stock andMoneyMarketModels 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

7.4.2 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Time Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8. Option Pricing 173
8.1 European Options in theBinomialTreeModel 174
8.1.1 OneStep 174
8.1.2 TwoSteps 176
8.1.3 General N-Step Model 178
8.1.4 Cox–Ross–RubinsteinFormula 180
8.2 AmericanOptionsin theBinomialTreeModel 181
8.3 Black–ScholesFormula 185
9. Financial Engineering 191
9.1 HedgingOptionPositions 192
9.1.1 DeltaHedging 192
9.1.2 GreekParameters 197
9.1.3 Applications 199
9.2 HedgingBusinessRisk 201
9.2.1 Valueat Risk 202
9.2.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.3 SpeculatingwithDerivatives 208
9.3.1 Tools 208
9.3.2 CaseStudy 209
10. Variable Interest Rates 215
10.1 Maturity-IndependentYields 216
10.1.1 InvestmentinSingleBonds 217
10.1.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.1.3 PortfoliosofBonds 224
10.1.4 Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.2 GeneralTermStructure 229
10.2.1 ForwardRates 231
10.2.2 Money MarketAccount 235

return,orbrieflyreturn:
K
S
=
S(1) − S(0)
S(0)
,
which is also uncertain. The dynamics of stock prices will be discussed in Chap-
ter 3.
The risk-free position can be described as the amount held in a bank ac-
count. As an alternative to keeping money in a bank, investors may choose to
invest in bonds. The price of one bond at time t will be denoted by A(t). The
1
2 Mathematics for Finance
current bond price A(0) is known to all investors, just like the current stock
price. However, in contrast to stock, the price A(1) the bond will fetch at time 1
is also known with certainty. For example, A(1) may be a payment guaranteed
by the institution issuing bonds, in which case the bond is said to mature at
time 1 with face value A(1). The return on bonds is defined in a similar way
as that on stock,
K
A
=
A(1) − A(0)
A(0)
.
Chapters 2, 10 and 11 give a detailed exposition of risk-free assets.
Our task is to build a mathematical model of a market of financial securi-
ties. A crucial first stage is concerned with the properties of the mathematical
objects involved. This is done below by specifying a number of assumptions,

S
and K
V
.The
return K
A
on a risk-free investment is deterministic.
Example 1.1
Let A(0) = 100 and A(1) = 110 dollars. Then the return on an investment in
bonds will be
K
A
=0.10,
that is, 10%. Also, let S(0) = 50 dollars and suppose that the random variable
S(1) can take two values,
S(1) =

52 with probability p,
48 with probability 1 − p,
for a certain 0 <p<1. The return on stock will then be
K
S
=

0.04 if stock goes up,
−0.04 if stock goes down,
that is, 4% or −4%.
Example 1.2
Given the bond and stock prices in Example 1.1, the value at time 0 of a
portfolio with x = 20 stock shares and y = 10 bonds is

1, 160 if stock goes up,
1, 040 if stock goes down.
What is the value of this portfolio at time 0?
It is mathematically convenient and not too far from reality to allow arbi-
trary real numbers, including negative ones and fractions, to represent the risky
and risk-free positions x and y in a portfolio. This is reflected in the following
assumption, which imposes no restrictions as far as the trading positions are
concerned.
Assumption 1.3 (Divisibility, Liquidity and Short Selling)
An investor may hold any number x and y of stock shares and bonds, whether
integer or fractional, negative, positive or zero. In general,
x, y ∈ R.
The fact that one can hold a fraction of a share or bond is referred to
as divisibility. Almost perfect divisibility is achieved in real world dealings
whenever the volume of transactions is large as compared to the unit prices.
The fact that no bounds are imposed on x or y is related to another market
attribute known as liquidity. It means that any asset can be bought or sold on
demand at the market price in arbitrary quantities. This is clearly a mathe-
matical idealisation because in practice there exist restrictions on the volume
of trading.
If the number of securities of a particular kind held in a portfolio is pos-
itive, we say that the investor has a long position. Otherwise, we say that a
short position is taken or that the asset is shorted. A short position in risk-free
1. Introduction: A Simple Market Model 5
securities may involve issuing and selling bonds, but in practice the same fi-
nancial effect is more easily achieved by borrowing cash, the interest rate being
determined by the bond prices. Repaying the loan with interest is referred to
as closing the short position. A short position in stock can be realised by short
selling. This means that the investor borrows the stock, sells it, and uses the
proceeds to make some other investment. The owner of the stock keeps all the

=1.60 dollars to a pound.
If this were the case, the dealers would, in effect, be handing out free money.
An investor with no initial capital could realise a profit of d
A
− d
B
=0.02
dollars per each pound traded by taking simultaneously a short position with
dealer B and a long position with dealer A. The demand for their generous
services would quickly compel the dealers to adjust the exchange rates so that
this profitable opportunity would disappear.
Exercise 1.3
On 19 July 2002 dealer A in New York and dealer B in London used the
following rates to change currency, namely euros (EUR), British pounds
(GBP) and US dollars (USD):
dealer A buy sell
1.0000 EUR 1.0202 USD 1.0284 USD
1.0000 GBP 1.5718 USD 1.5844 USD
dealer B buy sell
1.0000 EUR 0.6324 GBP 0.6401 GBP
1.0000 USD 0.6299 GBP 0.6375 GBP
Spot a chance of a risk-free profit without initial investment.
The next example illustrates a situation when a risk-free profit could be
realised without initial investment in our simplified framework of a single time
step.
Example 1.3
Suppose that dealer A in New York offers to buy British pounds a year from
now at a rate d
A
=1.58 dollars to a pound, while dealer B in London would sell

principle did exist, we would say that an arbitrage opportunity was available.
Arbitrage opportunities rarely exist in practice. If and when they do, the
gains are typically extremely small as compared to the volume of transactions,
making them beyond the reach of small investors. In addition, they can be more
subtle than the examples above. Situations when the No-Arbitrage Principle is
violated are typically short-lived and difficult to spot. The activities of investors
(called arbitrageurs) pursuing arbitrage profits effectively make the market free
of arbitrage opportunities.
The exclusion of arbitrage in the mathematical model is close enough to
reality and turns out to be the most important and fruitful assumption. Ar-
guments based on the No-arbitrage Principle are the main tools of financial
mathematics.
1.3 One-Step Binomial Model
In this section we restrict ourselves to a very simple example, in which the
stock price S(1) takes only two values. Despite its simplicity, this situation is
sufficiently interesting to convey the flavour of the theory to be developed later
on.
8 Mathematics for Finance
Example 1.4
Suppose that S(0) = 100 dollars and S(1) can take two values,
S(1) =

125 with probability p,
105 with probability 1 − p,
where 0 <p<1, while the bond prices are A(0) = 100 and A(1) = 110 dollars.
Thus, the return K
S
on stock will be 25% if stock goes up, or 5% if stock goes
down. (Observe that both stock prices at time 1 happen to be higher than that
at time 0; ‘going up’ or ‘down’ is relative to the other price at time 1.) The

We shall assume for simplicity that S(0) = A(0) = 100 dollars. Suppose that
A(1) ≤ S
d
. In this case, at time 0:
• Borrow $100 risk-free.
• Buy one share of stock for $100.
1. Introduction: A Simple Market Model 9
This way, you will be holding a portfolio (x, y) with x = 1 shares of stock
and y = −1 bonds. The time 0 value of this portfolio is
V (0) = 0.
At time 1 the value will become
V (1) =

S
u
− A(1) if stock goes up,
S
d
− A(1) if stock goes down.
If A(1) ≤ S
d
, then the first of these two possible values is strictly positive,
while the other one is non-negative, that is, V (1) is a non-negative random
variable such that V (1) > 0 with probability p>0. The portfolio provides an
arbitrage opportunity, violating the No-Arbitrage Principle.
Now suppose that A(1) ≥ S
u
. If this is the case, then at time 0:
• Sell short one share for $100.
• Invest $100 risk-free.


11, 600 if stock goes up,
9, 600 if stock goes down,
K
V
=

0.16 if stock goes up,
−0.04 if stock goes down.
The expected return, that is, the mathematical expectation of the return on the
portfolio is
E(K
V
)=0.16 × 0.8 − 0.04 × 0.2=0.12,
that is, 12%. The risk of this investment is defined to be the standard deviation
of the random variable K
V
:
σ
V
=

(0.16 − 0.12)
2
× 0.8+(−0.04 − 0.12)
2
× 0.2=0.08,
that is 8%. Let us compare this with investments in just one type of security.
If x =0, then y = 100, that is, the whole amount is invested risk-free. In
this case the return is known with certainty to be K

1.5 Forward Contracts
A forward contract is an agreement to buy or sell a risky asset at a specified
future time, known as the delivery date,forapriceF fixed at the present
moment, called the forward price. An investor who agrees to buy the asset is
said to enter into a long forward contract or to take a long forward position.If
an investor agrees to sell the asset, we speak of a short forward contract or a
short forward position. No money is paid at the time when a forward contract
is exchanged.
Example 1.5
Suppose that the forward price is $80. If the market price of the asset turns out
to be $84 on the delivery date, then the holder of a long forward contract will
buy the asset for $80 and can sell it immediately for $84, cashing the difference
of $4. On the other hand, the party holding a short forward position will have
to sell the asset for $80, suffering a loss of $4. However, if the market price of
the asset turns out to be $75 on the delivery date, then the party holding a
long forward position will have to buy the asset for $80, suffering a loss of $5.
Meanwhile, the party holding a short position will gain $5 by selling the asset
above its market price. In either case the loss of one party is the gain of the
other.
In general, the party holding a long forward contract with delivery date 1
will benefit if the future asset price S(1) rises above the forward price F .If
the asset price S(1) falls below the forward price F , then the holder of a long
forward contract will suffer a loss. In general, the payoff for a long forward
position is S(1) − F (which can be positive, negative or zero). For a short
forward position the payoff is F − S(1).
Apart from stock and bonds, a portfolio held by an investor may contain
forward contracts, in which case it will be described by a triple (x, y, z). Here
x and y are the numbers of stock shares and bonds, as before, and z is the
number of forward contracts (positive for a long forward position and negative
for a short position). Because no payment is due when a forward contract is

2
, −1) consisting of stock, a risk-free position, and
a short forward contract has initial value V (0) = 0. Then, at time 1:
• Close the short forward position by selling the asset for F dollars.
• Close the risk-free position by paying
1
2
× 110 = 55 dollars.
The final value of the portfolio, V (1) = F − 55 > 0, will be your arbitrage
profit, violating the No-Arbitrage Principle.
On the other hand, if F<55, then at time 0:
• Sell short the asset for $50.
• Invest this amount risk-free.
• Take a long forward position in stock with forward price F dollars and
delivery date 1.
The initial value of this portfolio (−1,
1
2
, 1) is also V (0) = 0. Subsequently, at
time 1:
1. Introduction: A Simple Market Model 13
• Cash $55 from the risk-free investment.
• Buy the asset for F dollars, closing the long forward position, and return
the asset to the owner.
Your arbitrage profit will be V (1) = 55 − F>0, which once again violates
the No-Arbitrage Principle. It follows that the forward price must be F =55
dollars.
Exercise 1.5
Let A(0) = 100, A(1) = 112 and S(0) = 34 dollars. Is it possible to
find an arbitrage opportunity if the forward price of stock is F =38.60

random variable
C(1) =

20 if stock goes up,
0 if stock goes down.
Meanwhile, C(0) will denote the value of the option at time 0, that is, the price
for which the option can be bought or sold today.
Remark 1.1
At first sight a call option may resemble a long forward position. Both involve
buying an asset at a future date for a price fixed in advance. An essential
difference is that the holder of a long forward contract is committed to buying
the asset for the fixed price, whereas the owner of a call option has the right
but no obligation to do so. Another difference is that an investor will need to
pay to purchase a call option, whereas no payment is due when exchanging a
forward contract.
In a market in which options are available, it is possible to invest in a
portfolio (x, y, z) consisting of x shares of stock, y bonds and z options. The
time 0 value of such a portfolio is
V (0) = xS(0) + yA(0) + zC(0).
At time 1 it will be worth
V (1) = xS(1) + yA(1) + zC(1).
Just like in the case of portfolios containing forward contracts, Assumptions 1.1
to 1.5 and the No-Arbitrage Principle can be extended to portfolios consisting
of stock, bonds and options.
Our task will be to find the time 0 price C(0) of the call option consistent
with the assumptions about the market and, in particular, with the absence of
arbitrage opportunities. Because the holder of a call option has a certain right,
but never an obligation, it is reasonable to expect that C(0) will be positive:
one needs to pay a premium to acquire this right. We shall see that the option
price C(0) can be found in two steps: