# MTH 6154 Queen Mary University of London Mathematics Interest Rates Quiz

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MTH6154: Financial Mathematics 1
Dr Dr Dudley Stark
Term 1, 2021-22
Contents
Preface
2
1 Interest rates and present value analysis
1.1 Interest rates . . . . . . . . . . . . . . .
1.2 Variable interest rates . . . . . . . . . .
1.2.1 The instantaneous interest rate .
1.2.2 The yield curve . . . . . . . . . .
1.3 Inflation . . . . . . . . . . . . . . . . .
1.4 Present value analysis . . . . . . . . . .
1.4.1 Defining the present value . . . .
1.4.2 Discount Factors . . . . . . . . .
1.4.3 Balancing present values . . . . .
1.5 Rates of return . . . . . . . . . . . . . .
1.5.1 The annualised rate of return and
1.5.2 Internal rate of return . . . . . .
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equivalent effective interest rate
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2 Immunisation of assets and liabilities
2.1 Measuring the effect of varying interest rates . . .
2.2 Reddington immunisation . . . . . . . . . . . . .
2.3 Immunisation in practice . . . . . . . . . . . . . .
2.4 Immunisation when interest is compounded yearly
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3 Bonds and the term structure of interest rates
3.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The present value of a bond . . . . . . . . . . . . . .
3.1.2 The no-arbitrage principle and the fair price of a bond
3.1.3 The bond yield . . . . . . . . . . . . . . . . . . . . .
3.2 The term structure of interest rates . . . . . . . . . . . . . .
3.2.1 Spot rates . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Forward rates . . . . . . . . . . . . . . . . . . . . .
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4 Stochastic interest rates
4.1 A fixed interest rate model . . . . . . .
4.2 A varying interest rate model . . . . .
4.2.1 The growth of a single deposit
4.2.2 The growth of regular deposits
4.3 Log-normally distributed interest rates .
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5 Equities and their derivatives
5.1 Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Pricing options: model-independent pricing principles . . . . . . . . . . . . . .
5.5 Pricing options via the risk-neutral distribution . . . . . . . . . . . . . . . . .
5.5.1 Betting strategies, the risk-neutral distribution, and the arbitrage theorem
5.5.2 Options pricing using the risk-neutral distribution . . . . . . . . . . . .
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6 The binomial model
6.1 The single-period binomial model . . . . . . . . . . . . . . . . . .
6.1.1 The no-arbitrage condition and the risk-neutral distribution
6.1.2 Option pricing in the single-period binomial model . . . . .
6.1.3 Replicating portfolios and the delta-hedging formula . . . .
6.2 The two-period binomial model . . . . . . . . . . . . . . . . . . .
6.2.1 Options pricing in the two-period binomial model . . . . . .
6.2.2 Replicating portfolios in the two-period binomial model . .
6.2.3 Option pricing via the two-period risk-neutral distribution .
6.3 The multi-period binomial model . . . . . . . . . . . . . . . . . .
6.3.1 Options pricing in the multi-period binomial model . . . . .
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7 From the multi-period binomial model to the Black-Scholes formula
7.1 General discrete-time models . . . . . . . . . . . . . . . . . . . . . .
7.2 The log-normal process . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 The Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Properties of the Black-Scholes formula . . . . . . . . . . . . .
7.4 The log-normal process as an approximation of the bin. model . . . . .
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Preface
This course is an introduction to financial mathematics, and is the first in a series of three
modules (along with Financial Mathematics II and Mathematical Tools for Asset Management1 ) that together give a thorough grounding in the theory and practice of modern
financial mathematics. While this course primarily deals with financial mathematics in discrete
time, FMII will cover financial mathematics in continuous time, and MTAM will focus on
risk management and portfolio theory.
So what exactly is financial mathematics? A simple definition would be that it is the use
of mathematical techniques to model and price financial instruments. These financial
instruments include:
• Debt (also known as ‘fixed-income’), such as cash, bank deposits, loans and bonds;
• Equity, such as shares/stock in a company; and
• Derivatives, including forwards, options and swaps.
As this definition suggests, there are two central questions in financial mathematics:
1
Formally known as Financial Mathematics III.
2
1. What is a financial instrument worth? To answer this, we will develop principled ways
in which to:
• Compare the value of money at different moments in time; and
• Take into account the uncertainty in the future value of an asset.
2. How can financial markets usefully be modelled? Note that modelling always involves a
This course introduces you to the basics of financial mathematics, including:
• An overview of financial instruments, including bonds, forwards and options.
• An analysis of the fundamental ideas behind the rational pricing of financial instruments,
including the central ideas of the time-value of money and the no-arbitrage principle. These concepts are model-free, meaning that they apply irrespective of the chosen
method of modelling the financial market;
• An introduction to financial modelling, including the use of the binomial model to
price European and American options.
The course culminates in a derivation, via an approximation procedure, of the celebrated
Black-Scholes formula for pricing European options.
3
1
Interest rates and present value analysis
The value of money is not constant over time – a thousand pounds today is typically worth
more than a contract guaranteeing £1,000 this time next year. To explain this, consider that
• If £1,000 was deposited into a bank account today, it would accumulate interest by this
time next year;
• Inflation may reduce the purchasing power of £1,000 over the course of the year.
In this chapter we explore how to take into account interest rates and inflation when comparing
assets that generate cash at different moments of time, using the concept of present value;
this gives us a way to make a principled comparisons between investments. Later in the course
we will use these ideas to assign a fair price to fixed-income securities,2 such as bonds.
An important theme in this chapter is the notation of standardisation, i.e. adjusting
interest rates and rates of return so that different rates can be compared fairly.
1.1
Interest rates
Suppose we borrow an amount P , called the principal, at nominal interest rate r. This
means that if we repay the loan after 1 year, we will need to repay the principal plus an extra
sum rP called the interest, so in total
P + rP = P (1 + r).
(1)
Similarly, if you put an amount P in the bank at an annualised interest rate r, then in a year’s
time the account value will grow to P (1 + r). Note that here we have assumed that the
nominal interest rate is annualised, meaning that interest is calculated once per year; unless
explicitly specified otherwise interest rates will always be assumed to be annualised
in this course.
Sometimes the interest is not calculated once per year, but instead is compounded every
of a year. This means that every n1 -th of a year you are charged (or, in the case of a bank
account, gain) interest at rate r/n on the principal as well as on the interest that has already
accumulated in previous periods. Continuing the example of the loan above, this would mean
that after one year we would owe

r n
.
(2)
P 1+
n
If the interest rate is ‘annualised’ then it is compounded annually (i.e. once per year), which
corresponds to n = 1 and gives the same result as in (1).
1
-th
n
Example 1.1. Suppose you borrow an amount P , to be repaid after one year at interest
rate r, compounded semi-annually. Then the following will happen sequentially throughout
the year. After half a year you will be charged interest at rate r/2, which is added on to the
principal. Thus, after 6 months you owe

r
.
P 1+
2
At the end of the year you are again charged interest at rate r/2, with the interest accumulating
on the entire sum owing, therefore at the end of the year you owe

r
r
r 2
P 1+
1+
=P 1+
.
2
2
2
2
A security is a tradable financial instrument.
4
This corresponds to choosing n = 2 in (2).
If n = 4 in (2) we say that interest is compounded quarterly, and if n = 12 we say that
interest is compounded monthly.
Example 1.2. Suppose that you borrow £100 at interest rate 18% compounded monthly.
After one year you owe
12

0.18
= 100 × 1.195 = £119.5.
100 1 +
12
In order to fairly compare interest rates we need to standardise to account for differences
in the compounding frequency. In this context the interest rate r is called the nominal rate.
To compare the effect of different compounding frequencies, we introduce the effective rate3
reff =
(amount after one year) − P
.
P
For an interest rate compounded every n1 -th of a year, we therefore have
reff =

P (1 + r/n)n − P
r n
= 1+
− 1.
P
n
(3)
The effective rate quantifies the net effect of interest over the course of a year, and so is a
fairer measure of the true effect of interest. Under UK consumer law, all credit providers must
publish their effective rate.
Example 1.3. Suppose that you borrow £100 at nominal interest rate 18% compounded
monthly. What is the effective rate?
Solution. We know from Example 1.2 that after one year you owe £119.5. Hence
reff =
119.5 − 100
= 0.195 = 19.5%.
100
Example 1.4. Credit card company A charges a 24.5% nominal rate compounded daily
whereas credit card company B charges a 24.7% nominal rate compounded monthly. Which
company offers the better deal?
Solution. The nominal interest rates do not give a fair comparison, since they are compounded
at different frequencies. A fairer comparison is given by the effective rates for both deals:
• For company A, the effective rate is

365
1
reff = 1 +
24.5%
− 1 = 0.2775.
365
• For company B, the effective rate is:

12
1
reff = 1 + 24.7%
− 1 = 0.2770.
12
3
Effective rates are sometimes also called ‘annual percentage rates’ (APRs).
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We conclude that company B offers the better deal even though the nominal rates advertised
might suggest the opposite.
Imagine now that interest is compounded at very small intervals, i.e. interest is compounded
every n1 -th of a year for n very large. In the limiting case, i.e. as n → ∞, we say that interest
is continuously-compounded, and in this case the amount owed after one year is

r n
lim P 1 +
= P er
n→∞
n
(we prove this rigorously in Coursework 1). The effective rate for continuously-compounded
interest at nominal rate r is therefore
reff =
P er − P
= er − 1.
P
Example 1.5. Suppose that a bank offers continuously-compounded interest at nominal
rate 5%. The effective rate is
reff = e0.05 − 1 = 0.05127 = 5.127%.
If a loan is taken out for T years, then the total amount owed after T years can be
calculated using the effective rate reff as
P (1 + reff ) × · · · × (1 + reff ) = P (1 + reff )T .
|
{z
}
T times
For example, if interest is compounded n times a year, this is

r nT
P 1+
,
n
and if it is compounded continuously
P (er )T = P erT .
Example 1.6 (The doubling rule). If a bank offers continuously-compounded interest at
nominal rate r, how long does it take for the amount of money in the bank to double?
Solution. Let T denote the time in years by the amount P doubles. Then
P erT = 2P

erT = 2

rT = ln 2,
and therefore
ln 2
0.693

.
r
r
For example, if the nominal rate is r = 12%, it takes 5.78 years for the amount to double.
T =
Remark 1.7. In investment folklore, the doubling rule is sometimes referred to as the ‘Rule
of 72’, since you can (roughly!) approximate ln 2 = 0.693 by 0.72, which often makes the
fraction easier to calculate, i.e. in the example above, 0.72/0.12 = 72/12 = 6 years. This
‘rule’ first appeared in print in 1494, long before logarithms were invented.
1.2
Variable interest rates
Interest rates may not be constant over time. To describe variable interest rates we introduce
the concepts of instantaneous interest rate and yield curve.
6
1.2.1
The instantaneous interest rate
Suppose that the interest rate at time t is equal to r = r(t); we call r(t) the instantaneous interest rate. We always assume that the instantaneous interest is continuouslycompounded, which means that for small h (and as long as r(t) is continuous at t), the
amount of interest that accrues between times t and t + h is equal to
1
er(t)h = 1 + r(t)h + r(t)2 h2 + . . . ≈ 1 + r(t)h.
2
We wish to calculate the amount P (t) that accumulates by time t, given that the amount
P (0) is deposited (or loaned) at time 0. It turns out that P (t) satisfies a simple differential
equation:
Theorem 1.8. Suppose that r(t) is a piece-wise continuous function. Then
P 0 (t) = P (t)r(t).
Proof. Let h denote a small time-period. Since interest is continuously compounded, the
interest accruing between times t and t + h is approximately equal to r(t)h, i.e.,
P (t + h) ≈ P (t) + P (t)r(t)h,
or equivalently
P (t + h) − P (t)
≈ P (t)r(t).
h
As h becomes smaller, the above approximation becomes more and more accurate, and so
P (t + h) − P (t)
= P (t)r(t).
h→0
h
lim
Recognising the left-hand side as the definition of the derivative, we conclude that
P 0 (t) = P (t)r(t),
The above equation is a separable first order differential equation, easily solved by writing
P 0 (t)
= r(t)
P (t)
and integrating both sides:
Z
0
Thus
u=t
[ln P (u)]u=0
Z
=
t
P 0 (u)
du =
P (u)
t
Z
r(u) du.
0
t
Z
r(u) du =⇒ ln P (t) = ln P (0) +
0
t
r(u) du,
0
and so
P (t) = P (0) exp
Z
t

r(u) du ,
(4)
0
which is the desired relation between the amount P (t) in your bank at time t and the timevarying interest rate function r(t).
Note that if r(t) = r is constant, then P (t) = P (0)ert . Thus, the general formula (4)
reduces to the familiar formula in the special case of constant interest rate.
7
1.2.2
The yield curve
Suppose that the instantaneous interest rate is r(t). The yield curve r̄(t) is defined to be
the average value of r(t) on the interval (0, t):
Z
1 t
r̄(t) =
r(u) du.
t 0
The yield curve allows us to write the formula for P (t) in (4) as
Z t

P (t) = P (0) exp
r(u) du = P (0)er̄(t)t .
0
The effective rate can also be expressed in terms of the yield curve. Since the interest
rate is variable, this rate reff (T ) depends on the period of time T the money is deposited/loaned, and is equal to
reff (T ) = er̄(T ) − 1.
Example 1.9. Suppose you deposit £100 in a bank offering the instantaneous interest rate
(
each year indefinitely. What is its present value if the interest rate is r?
Solution. The cash-flow stream generated by the perpetuity is a = (a, a, a, . . .), and so its
present value is
P V (a) =

X
i=1
a
1/(1 + r)
a
a
=a×
=
= .
i
(1 + r)
1 − 1/(1 + r)
(1 + r) − 1
r
11
Remark 1.17. Note that the present value ar of the perpetuity can also be justified by a
simple replication argument. Suppose that instead of purchasing a perpetuity you place ar
in a bank account. After one year you have ar (1 + r) in the account. Suppose you withdraw
a to replicate the payment of the perpetuity. Then you are left with
a
a
a
(1 + r) − a = + a − a = .
r
r
r
a
At the end of the second year, this will have grown to r (1 + r), so if you again withdraw a
you are left with
a
a
a
(1 + r) − a = + a − a = .
r
r
r
If you continue like this you can withdraw a from the bank account each year forever, which
replicates exactly the cash-flow stream generated by the perpetuity.
1.4.2
Discount Factors
It is often simpler to write the present value of a cash-flow stream using the concept of
discount factors. This is especially the case if interest rates vary over time.
Definition 1.18. The discount factor to T years, D(T ), is the present value of a payment
of 1 unit of cash in T years’ time. Equivalently, it is the reciprocal of what a bank account
holding 1 unit of cash now will be worth in T years’ time.
If interest rates are positive (which is most often the case) the discount factor is smaller
than 1 hence the phrase ‘discount’. This indicates that having money in the future is worth
less than having the same amount of money today.
Examples 1.19.
1. If the interest is compounded annually at nominal rate r then the discount factor to T
years is
1
.
D(T ) =
(1 + r)T
2. If the interest is compounded at a different frequency, we can write the discount factor
to T years in terms of the effective rate reff as
1
D(T ) =
.
(1 + reff )T
For example, if interest is compounded monthly then
1
D(T ) =
,
1 12T
1 + r 12
if interest is continuously-compounded then
D(T ) = e−rT .
3. If r(t) is an instantaneous interest rate then
D(T ) = e−r(T ) T .
Proposition 1.20. If a cash-flow stream consists of payments a1 , a2 , . . . , an transacted at
times t1 , t2 , . . . , tn , then the present value of this cash-flow stream is
PV =
n
X
i=1
12
D(ti )ai .
1.4.3
Balancing present values
In many real-life situations it is essential that the present values of assets (i.e. incoming
payments) and liabilities (i.e. outgoing payments) are equal. The following is a practical
example of such a situation:
Example 1.21. Suppose that Alex is self-employed and wants to save for her retirement. She
wants an income of £1,000 a month starting in 20 years’ time and lasting for 30 years. What
amount of money does she have to save every month (starting today) for the next 20 years to
fund her retirement? Suppose that the nominal rate is 6% compounded monthly.
There exist various possible solutions to this problem. Here are two of them:
Solution 1. Let a denote the as yet unknown amount of pounds she must deposit every month,
and let D denote the cash-flow stream generated by these deposits. Similarly, let w =
£1,000, and let W denote the cash-flow stream generated by her monthly withdrawals for her
retirement. In order to fund her retirement, the present value of her deposits must equal the
present value of her withdrawals, i.e.
P V (D) = P V (W ).
We first calculate P V (D). It is convenient to work with the monthly discount factor
β = D(1/12) =
1
1
.
0.06 =
1.005
1 + 12
Since there are 240 months in 20 years,
P V (D) = a + aβ + aβ 2 + · · · + aβ 239 = a(1 + β + β 2 + · · · + β 239 ) = a
1 − β 240
.
1−β
We next calculate P V (W ). Since her first withdrawal will take place in 240 months’ time,
and since there are 360 months in 30 years,
P V (W ) = wβ 240 + wβ 241 + · · · + wβ 240+359 = wβ 240 (1 + β + · · · + β 359 ) = wβ 240
1 − β 360
.
1−β
Since P V (D) = P V (W ), we conclude that
a
1 − β 240
1 − β 360
= wβ 240
1−β
1−β

a = wβ 240
1 − β 360
= £361
1 − β 240
that is, she has to deposit £361 every month to fund her retirement.
Solution 2. A alternative approach is to compare the value of the cash-flow streams at the
end of year 20, instead of at the present time. Using the notation from the previous solution,
the value of her deposits at the end of year 20 is
aβ −240 + aβ −239 + … + aβ −2 + aβ −1 = aβ −1
β −240 − 1
β −1 − 1
since β −1 is the amount that £1 accumulates in a deposit of one month. Similarly, the value
at the end of year 20 of all her retirement withdrawals is
w + wβ + wβ 2 + . . . + wβ 359 = w
1 − β 360
.
1−β
Equating these values, we have
a β −1
β −240 − 1
1 − β 360
=
w
β −1 − 1
1−β

13
a=w
1 − β 360
= £360.9 .
β −240 − 1
1.5
Rates of return
Suppose you invest an amount P (the ‘principal’) which at a later times results in a value
of Pfinal . How can you assess the quality of this investment? Three possible quantifiers are:
• The return on the investment is the quantity Pfinal .
• The profit on the investment is the quantity Pfinal − P .
• The rate of return on the investment is the profit as a percentage of the principle
R=
Pfinal − P
.
P
(5)
Remark 1.22. Be careful when using the phrases ‘profit’ and ‘return’; practitioners are not
always consistent in their use.
Examples 1.23.
1. Suppose we invested P = £150,000 in a property which is now valued at Pfinal =
£200,000. What is the rate of return? This is given by
R=
200,000 − 150,000
= 0.333 = 33.3%.
150,000
2. What is the total return on an investment of \$12,000 that guarantees a 20% rate of
return? What is the profit? Rearranging (5), the return is
Pfinal = P (1 + R) = \$12,000 × (1 + 0.2) = \$14,400
and the profit is
P × R = \$12,000 × 0.2 = \$2,400(= \$14,400 − \$12,000).
1.5.1
The annualised rate of return and the equivalent effective interest rate
The rate of return does not fairly evaluate the performance of an investment, since it does
not take into account how much time it took for the investment to accrue its value.
For example, a rate of a return of 33.3% as in Example 1.23 would be spectacular if gained
in two weeks, but would be less spectacular if the increase took 40 years. To fairly compare
rates of return, we need to standardise them to account for differences in time-period.
A naive standardisation would simply divide the rate of return R by the time T it took for
the investment to accrue its value, i.e.
R
r=
T
called annualised rate of return. A better standardisation is to consider the equivalent
effective interest rate r that a bank deposit would need to offer in order to match the return
on the investment. This is defined to satisfy
P (1 + r)T = Pfinal = P (1 + R).
Rearranging, this gives
r = (1 + R)
1/T
−1=
14
P
final
P
1/T
− 1.
Examples 1.24.
1. Suppose we invested £1,500 in shares two year’s ago, and these are now worth £1,750.
The equivalent effective interest rate is
P 1/T
1750 1/2
final
r=
−1=
− 1 = 0.0801 = 8.01%, .
P
1500
2. How much profit do we earn if we invest \$2,200 for two years in an investment that
guarantees an equivalent effective interest rate of 3%? We have
Pfinal = P (1 + r)T = \$2,200(1 + 0.03)2 = \$2,330
and so the profit is
Pfinal − P = \$2,330 − \$2,200 = \$130.
1.5.2
Internal rate of return
When considered the performance of an investment that generates a cash-flow stream with
payments at multiple instances of time, we need a measure of the annualised rate of return
that takes into account all the payments. This is the internal rate of return (IRR).
Suppose that you invest amount P and after T years you get back amount Pfinal . Recall
that the equivalent effective interest rate of this investment satisfies
Pfinal = P (1 + r)T

P =
Pfinal
.
(1 + r)T
More generally, suppose the initial investment P generates the cash-flow stream a = (a1 , a2 , . . . , an )
with payments at times t1 , t2 , . . . , tn . Then we define the internal rate of return to be the
number r ∈ (−1, ∞) such that
n
X
ai
,
P =
ti
(1
+
r)
i=1
that is, the solution of the equation f (r) = 0, for the function
f (r) = −P +
n
X
ai (1 + r)−ti .
i=1
The internal rate of return is the ‘hypothetical’ effective interest rate that makes the present
value of the cash-flow stream equal to the principal of the investment. In other words, it is
the effective interest rate that a bank would need to offer in order to replicate the payment
stream of the investment. An investment can be considered to be ‘good’ if the IRR exceeds
the effective interest rate offered in the market, and ‘bad’ if the IRR is less than the effective
interest rate offered in the market.
The definition of the IRR is only meaningful if the equation f (r) = 0 has a unique solution
r ∈ (−1, ∞). We next proves this to be true (at least, for positive cash-flow streams):
Theorem 1.25. Suppose P > 0 and ai , ti > 0 for i = 1, . . . , n. Then the equation
f (r) = 0

P =
n
X
i=1
has a unique solution in (−1, ∞).
15
ai (1 + r)−ti
(6)
Proof. Observe the following properties of the function f on (−1, ∞):
• f is strictly decreasing, since
f 0 (r) =
n
X
ai
i=1
−ti
< 0. (1 + r)ti +1 • limr→−1 f (r) = +∞ since lim f (r) = lim r→−1 r→−1 −P + n X −ti ai (1 + r) = −P + n X i=1 ai lim (1 + r)−ti = +∞. i=1 r→−1 • limr→+∞ f (r) = −P < 0 since lim f (r) = lim r→+∞ r→+∞ −P + n X −ti ai (1 + r) = −P + i=1 n X i=1 ai lim (1 + r)−ti = −P. r→+∞ Since f is continuous on (−1, ∞) and strictly decreasing from +∞ to −P < 0, f attains the value 0 at exactly one point in r ∈ (−1, ∞). Remark 1.26. One by-product of the proof of Theorem 1.25 is that the present value of a cash-flow stream is a decreasing function of the interest rate r. We will study this fact in more detail in the next chapter. In general calculating the IRR is difficult since it requires solving equation (6), which is a polynomial equation of degree n for which no closed solutions are available in general.5 However the case n = 2 can be solved easily. Example 1.27. Suppose that an investor bought shares in a company for £1,000 at the beginning of 2015, and sold them at the beginning of 2017 for £1,050. At the end of 2015 and 2016 the investor also received dividends (i.e. cash payments) of £10 and £20, respectively. (a) What is the IRR of this investment? (b) What is the IRR of this investment if the investor had to pay 10% tax on the dividends and 18% capital gains tax on the profit from the increase in share price? (c) Suppose the nominal interest rate is 3.5%. By comparing the IRR to the nominal interest rate, judge whether the purchase of these shares was a good investment. Solution. (a) The IRR is the solution in r ∈ (−1, ∞) of the equation 1,000 = 20 1,050 10 1,070 10 + + = + . (1 + r) (1 + r)2 (1 + r)2 (1 + r) (1 + r)2 Rewriting this, we have 1,000(1 + r)2 − 10(1 + r) − 1,070 = 0, 5 Instead such equations must be solved numerically using trial and error, or sophisticated variations of this. 16 which is a quadratic equation in (1 + r) with solutions 1.0394 and − 1.0294. This implies that the solutions r are 0.0394 and − 2.0294 (which we reject, since r > −1).
Thus the IRR of this investment is r = 3.94%.
(b) Since the two dividend payments were taxed at 10%, the investor received only 10 − 1 =
£9 at the end of 2015 and 20−2 = £18 at the end of 2016. As the investor made a profit
of £1,050 − £1,000 = £50 after selling the shares, they also had to pay 0.18 · 50 = £9
capital gains tax. This reduces the investor’s return to £1,050 − £9 = £1,041.
The IRR in this case is the solution in r ∈ (−1, ∞) of the equation
1,000 =
9
18 + 1,041
+
.
(1 + r)
(1 + r)2
Again, this is a quadratic equation in r with solutions
0.0335
and
− 2.0245 (which we reject, since r > −1).
Thus, in this case the IRR of this investment is r = 3.35%. Note that this is smaller
than the rate of return calculated in (a), which reflects the fact that the tax reduced
the profitability of the investment.
(c) Ignoring taxes (as in part (a)), the IRR exceeded the nominal interest rate, so the
purchase of the shares was a good investment. However, taking into account taxes, the
IRR was lower than the nominal interest rate (as in part (b)), so overall the purchase of
the shares was not a good investment.
17
2
Immunisation of assets and liabilities
A company that expects to receive payments in the future (or needs to make payments in
the future) is exposed to fluctuations in the interest rate. In this chapter we explore how the
cash-flow of a company can be structured in a way that minimises this risk – this is referred
to as the immunisation of assets and liabilities.
Manipulating a cash-flow stream so as to minimise exposure to fluctuations in financial
markets more generally is called risk management, and is the focus of Mathematical
Tools for Asset Management. The action of entering into transactions for the purpose of
managing risk is referred to as hedging.
2.1
Measuring the effect of varying interest rates
Let us first consider how the present value of a cash-flow stream varies when the interest rate
r varies. Let r denote the current interest rate (which may or may not be compounded), and
let D(T ) be the discount factor to T years (sometimes we write D(T ; r) instead of D(T ) to
stress the dependence on r). Suppose that the cash-flow stream consists of a sequence of
payments Ci transacted at times ti . The present value V (r) of this cash-flow stream is then
V (r) =
m
X
Ci D(ti ; r).
i=1
We have already observed in Chapter 1.5.2 that V (r) is a decreasing function of the interest
rate r (at least, if all payments Ci are positive). We can analyse how rapidly V (r) decreases
with r by considering derivatives of V (r).
First-order properties. A simple way to quantify how much V (r) decreases with r is to
compute V 0 (r). For technical reasons explained below, practitioners prefer to use the following
slightly different quantity:
Definition 2.1. If V (r) > 0, the effective duration of the cash-flow stream is defined as
ν=−
V 0 (r)
V (r)
Remarks 2.2. Let us explain why ν is preferred to V 0 (r):
1. The derivative V 0 (r) is measured in the same monetary units (e.g. pounds, dollars, pence
etc.) as the present value V (r). By dividing V 0 (r) by V (r), we standardise V 0 (r) to
remove the monetary units. In fact, the units in which ν is measured is years (why?).
2. Since V (r) is decreasing in r for a positive cash-flow stream, the minus sign is included
to ensure that ν is positive for such a stream (although it can be negative if the cash-flow
stream has negative payments).
Why is the effective duration called a ‘duration’ ? To answer this, let us first define a
second quantity, called simply the ‘duration’:
Definition 2.3. If V (r) > 0, the duration of the cash-flow stream is defined as
Pm
Pm
ti Ci D(ti )
i=1 ti Ci D(ti )
τ=
= Pi=1
.
m
V (r)
i=1 Ci D(ti )
18
In the case of positive payments Ci > 0, τ can be considered a weighted average of the
times ti
ω1 t1 + · · · + ωm tm
τ=
ω1 + · · · + ωm
where the weights ωi = Ci D(ti ) > 0 are the present value of the payments.
What is the relationship between the effective duration ν and the duration τ ? In the case
of continuously-compounded interest, it turns out that they are exactly the same thing.
Proposition 2.4. If interest is continuously-compounded, then the effective duration is equal
to the duration, that is, ν = τ .
Proof. The discount factors are D(ti ) = e−rti . Calculating
V (r) =
m
X
Ci e
−rti
and
0
V (r) =
i=1
we have
V 0 (r)
=−
ν=−
V (r)
m
X
−ti Ci e−rti ,
i=1
Pm
i=1
−ti Ci e−rti
=
V (r)
Pm
i=1 ti Ci D(ti )
V (r)
= τ.
More generally, the duration is equal to the effective duration up to a constant.
Proposition 2.5. There is a constant c0 , depending on the interest rate but not on the
cash-flow stream, such that
ν = c0 τ.
Remark 2.6. As we have seen, in the case of continuously-compounded interest the constant
c0 = 1. The value of c0 in the general case is more complex (and not very important), but if
payments are made yearly and interest is compounded n times a year then (prove this!)
ν = (1 + r/n)−1 τ.
The equivalence between the effective duration and the duration leads to the following
useful general principle:
If cash-flow stream A has a larger duration than cash-flow stream B, then the
present value of A decreases in value faster than the present value of B when the
interest rate increases.
Remark 2.7. Whether it is ‘better’ to have a cash-flow stream with large duration depends
on the context, in particular whether the cash-flow stream consists of assets or liabilities and
whether the interest rate increases or decreases.
Example 2.8. You expect to be paid £1,000 at the end of this year and the end of the
following year, and need to make a payment of £2,215 in three years’ time. The interest rate
is currently 7% and is compounded yearly.
(a) Verify that the present value of your assets (incoming payments) and your liability
(outgoing payment) are equal.
(b) If the interest rate were to increase to 8%, would you be in a better or worse financial
position (in terms of the net present value of assets and liabilities)?
19
Solution.
(a) The discount factor to one year is β = 1/1.07. The present value of the assets is
1,000β + 1,000β 2 = 1,808,
whereas the present value of the liability is
2,215β 3 = 1,808.
(b) The present value of both the assets and liability will decrease if the interest rate increases
from 7% to 8%, so whether you are in a better financial position depends on which
decreases more steeply. This can be answered by comparing durations.
Calculating the durations explicitly gives
1 × 1000β + 2 × 1000β 2
= 1.483
τ (A) =
1808
and
3 × 2215β 3
τ (L) =
=3
2215β 3
Hence the duration of the liability is larger than the duration of the assets, meaning
that the present value of the liabilities will decrease more than the present value of the
assets, so the increase in interest rates would put you in a better financial position.
Note that we actually didn’t have to do the calculations explicitly: as a weighted average
of payment times, the duration of the assets must be between 1 and 2, whereas the
duration of the liability is equal to 3.
Second-order properties. We can refine our analysis of how V (r) varies with r by considering V 00 (r). As for the effective duration, it is more natural to considered the normalised
version V 00 (r)/V (r), which is called the convexity.
Definition 2.9. If V (r) > 0, the convexity of the cash-flow stream is defined as
c=
V 00 (r)
.
V (r)
Proposition 2.10. If interest is continuously-compounded, then the convexity is
Pm 2
Pm 2 −rti
ti Ci e
i=1 ti Ci D(ti )
c=
= Pi=1
.
m
−rti
V (r)
i=1 Ci e
(7)
Again, in the case of positive payments Ci > 0, the convexity c can be interpreted as a
‘weighted average’ of the squared payment times t21 , . . . , t2n . However, this does not have a
direct financial meaning.
Remark 2.11. In general, the relationship between the convexity c and the weighted average
in (7) is more complex. For instance, if interest is compounded n times a year then
Pm
ti (ti + 1/n)Ci D(ti )
× (1 + r/n)−2 .
c = i=1
V (r)
20
2.2
Reddington immunisation
We now discuss how a company can minimise their exposure to fluctuations in the interest
rate. Let the current interest rate be r = r0 . We assume that a company has a stream
of positive future cash-flows (i.e. incoming payments), called assets, whose present value is
VA (r0 ), and a stream of negative future cash-flow (i.e. outgoing payments), called liabilities,
with present value −VL (r0 ). The present value of the combined cash-flow stream is therefore
V (r0 ) = VA (r0 ) − VL (r0 ).
The ‘immunisation’ of these cash-flow streams refers to a sequence of three conditions that
VA and VL need to satisfy in order to minimise expose to fluctuating interest rates; if all three
conditions are satisfied the cash-flows are said to be Reddington immune.
Zero-order immunisation: Matching assets and liabilities Zero-order immunisation
simply means that the present value of assets is matched by the present value of liabilities:
Definition 2.12. A cash-flow stream is said to be zero-order immune if the present value
of assets and liabilities is the same, i.e. VA (r0 ) = VL (r0 ).
Note that zero-order immunisation can in general only be achieved for the present value
of the interest rate r = r0 , and will fail as soon as r differs from r0 .
First-order immunisation: Matching the effective durations of assets and liabilities
Now assume that the cash-flow is zero-order immune, and consider the effect of a change in
the interest rate to r = r0 + ε, for ε a small number. As we previously observed, in general
VA (r0 + ε) 6= VL (r + ε), and we can use Taylor expansions to approximate the difference:
VA (r0 + ε) ≈ VA (r0 ) + VA0 (r0 )ε = VA (r0 ) − VA (r0 )νA ε = VA (r0 ) (1 − νA ε)
VL (r0 + ε) ≈ VL (r0 ) + VL0 (r0 )ε = VL (r0 ) − VL (r0 )νL ε = VL (r0 ) (1 − νL ε)
(8)
(9)
where νA = −VA0 (r0 )/VA (r0 ) and νL = −VL0 (r0 )/VL (r0 ) are the effective durations of the
cash-flow streams corresponding to the assets and liabilities. Hence
V (r0 + ε) = VA (r0 + ε) − VL (r + ε) ≈ VA (r0 ) (1 − νA ε) − VL (r0 ) (1 − νL ε) ,
which in the case of zero-order immunisation, simplifies to
V (r0 + ε) ≈ VA (r0 ) (νL − νA ) ε.
This quantity is zero if and only if the effective durations are equal, i.e. νA = νL . This is
equivalent to the durations being equal, i.e. τA = τB .
Definition 2.13. A cash-flow stream is said to be first-order immune if (i) it is immune to
zero-order, i.e. VA (r0 ) = VL (r0 ), and (ii) if the effective durations (equiv. durations) of the
cash-flow streams corresponding to the assets and liabilities are equal, i.e. if νA = νB (equiv.
τA = τB ).
If a cash-flow stream is first-order immune, then the value of V (r0 + ε) is constant (up to
small errors) for small . The relevant errors are very small since they correspond to terms of
order ε2 or higher, e.g., if ε = 0.1% = 0.001 then ε2 = 0.000001.
21
Reddington immunisation: Convexity We can use the notion of convexity to refine the
approximations in (8) and (9):
1
1
VA (r0 + ε) ≈ VA (r0 ) + VA0 (r0 )ε + VA00 (r0 )ε2 = VA (r0 ) − VA (r0 )νA ε + VA (r0 )cA ε2
2
2

1
= VA (r0 ) 1 − νA ε + cA ε2
2
1
1
VL (r0 + ε) ≈ VL (r0 ) + VL0 (r0 )ε + VL00 (r0 )ε2 = VL (r0 ) − VL (r0 )νL ε + VL (r0 )cL ε2
2
2

1
= VL (r0 ) 1 − νL ε + cL ε2
2
where cA = VA00 (r0 )/VA (r0 ) and cL = VL00 (r0 )/VL (r0 ) are the convexity of the cash-flow
streams corresponding to the assets and liabilities respectively. Hence in the case of first-order
immunisation,

1
1
V (r0 + ε) ≈ VA (r0 ) 1 − νA ε + cA ε2 − VL (r0 ) 1 − νL ε + cL ε2
2
2
1
= VA (r0 ) (cA − cL )ε2
2
Clearly it would be advantageous to demand that cA = cL so that V (r0 + ε) was constant for
small ε up to very small errors (of order ε3 ). In practice, it is often sufficient to demand that
cA ≥ cL , which ensures that, up to very small errors, V (r0 + ε) ≥ V (r0 ).
Definition 2.14. A cash-flow stream is Reddington immune if it is first-order immune, i.e.
VA (r0 ) = VL (r0 )
and νA (r0 ) = νL (r0 ),
and if, in addition, the convexities of the assets and liabilities satisfy cA ≥ cL .
If a cash-flow stream is Reddington immune, up to very small errors any change in the
interest rates will increase V (r) (or, at least, V (r) will not decrease)
2.3
Immunisation in practice
Let us see how immunisation might be applied in practice. For this we will make use of zerocoupon bonds, which are simply assets that pay a fixed amount (say £1) at the end of a
fixed number of years (bonds will be discussed in more detail in the next chapter).
Example 2.15. An investor has a liability of £20,000 to be paid in 4 years and another of
£18,000 to be paid in 6 years. Suppose the interest rate is r = 8% and is continuouslycompounded.
(a) Show that Reddington immunisation can be achieved by owning a combination of 2-year
and 7-year zero-coupon bonds.
(b) Explain why first-order immunisation cannot be achieved if the fund owns a combination
of 2-year and 3-year zero-coupon bonds.
Solution. (a) The yearly discount factor is β = e−0.08 = 0.923. Suppose we own 2-year
bonds paying £P and 7-year bonds paying £Q. Zero-order immunisation requires that
P β 2 + Qβ 7 = 20,000β 4 + 18,000β 6 = 25,661.
22
Moreover, first-order immunisation (i.e. τA = τL ) requires that
2P β 2 + 7Qβ 7 = 4 × 20,000β 4 + 6 × 18,000β 6 = 124,900.
Solving the two equations for P and Q gives the solution
P = 12,840,
Q = 25,770.
To determine whether Reddington immunisation is achieved for this solution, we calculate
the convexities. Since interest is continuously-compounded, we have
Pn 2
t Ci β ti
22 P β 2 + 72 Qβ 7
cA = i=1 i
=
= 29.8
VA (r)
25,661
and
Pn
cL =
2
ti
i=1 ti Ci β
VL (r)
=
42 × 20,000β 4 + 62 × 18,000β 6
= 24.7.
25,661
Since cA > cL , Reddington immunisation is achieved.
(b) The duration of a combination of 2-year and 3-year bonds is at most 3 years, which
cannot be equal to the duration of the liabilities (which is at least 4 years), so in this
case first-order immunisation is impossible.
2.4
Immunisation when interest is compounded yearly
If interest is compounded yearly at rate r, then the present value of a payment stream of
payments Ci made at times ti for i = 1, 2, . . . , n is
V (r) =
n
X
i=1
Ci
.
(1 + r)ti
Exactly the same definitions of ν and c can be made for this present value function as for
continuously compounded interest and the same immunisation conditions apply. However,
now
n
X
t i Ci
0
V (r) = −
(1 + r)ti +1
i=1
and
V 0 (r) =
n
X
ti (ti + 1)Ci
i=1
(1 + r)ti +2
.
With β defined as β = 1/(1 + r),
V (r) =
n
X
Ci β ti ,
i=1
0
V (r) = −
n
X
ti Ci β ti +1 ,
i=1
and
V 00 (r) =
n
X
ti (ti + 1)Ci β ti +2 ,
i=1
23
The duration is
Pn
ti Ci β ti
τ = Pi=1
n
ti
i=1 Ci β
while the effective duration is
V 0 (r)
=
ν=−
V (r)
Pn
ti +1
i=1 ti Ci β
P
.
n
ti
i=1 Ci β
As opposed to the case for continuously compounded interest, ν = βτ .
Example 2.16. A fund must make paments of £50, 000 at the end of the sixth and eigth
year from now. Show that immunisation can be acheived for 7% interest with a comination
of 5-year zero-coupon bonds and 10-year zero-coupond bonds.
Solution. Suppose we purchase 5-year bonds paying P and 10-year bonds paying Q. Then,
with β = (1.07)−1 ,
VA (0.07) = P β 5 + Qβ 10 .
We also have
VL (0.07) = 50, 000(β 6 + β 8 ) = 62, 418
and so zero order immunisation implies that
P β 5 + Qβ 10 = 62, 418.
(10)
The effective duration of the assets is
νA =
5P β 6 + 10Qβ 11
62418
while the effective duration fo the liabilities is
νL =
404398
50, 000(6β 7 + 8β 9 )
=
.
62418
62418
Setting νA = νL gives
5P β 6 + 10Qβ 11 = 404398.
(11)
Solving (10) and (11) for P and Q gives
P = £53, 710,
Q = £47, 454.
(12)
We must still show that the third immunisation condition is satisfied. The convexity of the
assets is
V 00 (0.07)
(5)(6)P β 7 + (10)(11)Qβ 12
cA = A
=
= 53.21
VA (0.07)
62418
while the convexity of the liabilities is
cL =
VL00 (0.07)
50000((6)(7)β 8 + (8)(9)β 10 )
=
= 48.90.
VL (0.07)
62418
Since cA ≥ cL , the third condition is satisfied and (12) gives immunisation.
24
3
Bonds and the term structure of interest rates
Bonds are an example of a debt or fixed-income security – these are tradable financial
instruments whose cash-flow is fixed or predetermined. Other examples include bank deposits,
loans, interest rate futures, interest rate swaps, swaptions, caps, floors, etc. (often credit
derivatives are also included, e.g., CDSs (credit default swaps), CDOs (collateralised debt
obligations), convertible bonds, etc., since their structure is similar). Note that, although the
cash-flow is predetermined, the present value of a fixed-income security can fluctuate, since
their value also depends on the prevailing interest rates.
Fixed-income securities can be contrasted with equity, such as shares and equity indices.
These are securities whose cash-flow is not pre-determined. Other financial instruments of
this type include FX (foreign exchange) and commodities, and collectively they are sometimes
known as variable-income securities.
In this chapter we will use present value analysis and no-arbitrage arguments to assign a
fair price to bonds. We also consider the term structure of bonds, which allow us to define
a generalised notion of interest rates that depend, not just on the present time, but also on
the period of time in which money is invested. Using this idea, we will see that all bond prices
and all interest rates can be expressed in terms of the price of zero-coupon bonds.
3.1
Bonds
A bond is a contract issued by a government or a company in order to raise capital. Investors
that purchase a bond are promised a stream of fixed payments known as coupons, plus an
extra payment at the bond expiry, known as the redemption.
A bond is described by:
• An issue price, which is the amount the investor pays for the bond. It is sometimes
determined by an auction.
• Its face value, also called the principal. This will determine the final redemption payment, and for some bonds will also determine the coupon values.
• The expiry (or ‘maturity’) of the bond. This is the date of the last coupon payment
and/or when the investor receives the redemption payment.
• The coupon value and frequency, which expresses the value and frequency of the
coupon payments. The value is sometimes expressed as a fixed amount, and sometimes
as a (annualised) coupon rate, which is a percentage of the face value.
If there are m coupon payments in a year then each of them will be equal to the annual
coupon divided by m. A zero-coupon bond is a bond with no coupon payments.
Note that the coupon rate is in general different to the interest rate.
• The redemption value, which expresses the value of the redemption payment at expiry.
This is again sometimes expressed as a fixed amount, and sometimes as a fraction
R of the face value. If R = 1, then the bond is said to be ‘redeemable at par’, if
R > 1, ‘redeemable above par’, and if R < 1, ‘redeemable below par’. Most bonds are redeemable at par. 25 The cash-flow stream generated by a typical bond might be as follows:6 R × Face value + Coupon Now Coupon Coupon Coupon 1Y 2Y Coupon Coupon 3Y 8Y 9Y 10Y Issue price Examples 3.1. 1. A 5-year bond with a face value of £100,000,000 and an annual coupon rate of 3% will pay 5 annual coupon payments of £3,000,000. Additionally, if the the bond is redeemable at par (R = 1), at expiry the investor will be paid back the face value of the bond: \$100,000,000. 2. What coupons are paid by a 10-year bond with face value \$150,000,000 and semi-annual coupon paid at an annual rate of 2%? The bond will pay 20 coupon payments every half-year for 10 years, each worth 1 \$150,000,000 × 2% × = \$1,500,000. 2 3.1.1 The present value of a bond The present value of a bond is simply equal to the present value of the cash-flow stream that the bond generates. By default we do not include the issue price in the cash-flow stream, unless specifically included (immediately after purchase, the initial payment is no longer relevant to the bond value). Example 3.2. Assuming an interest rate of r, the present value of a zero-coupon bond with face value 1 that is redeemable at par in T years is equal to 1 . (1 + r)T Example 3.3. Assuming an interest rate of r, the present value of a bond maturing in T years with yearly coupon payments of C, issue price P0 , face value P , and redemption price RP is 1 1 1 1 PV = C+ C + ··· + C+ RP. (13) 2 T 1+r (1 + r) (1 + r) (1 + r)T Example 3.4. Suppose the interest rate is 1.5%. The present value of a 3-year bond with face value £100,000, redeemable at par, with annual coupon payments of 1.7% is 0.017 × 100,000 0.017 × 100,000 0.017 × 100,000 100,000 PV = + + + 2 3 1 + 0.015 (1 + 0.015) (1 + 0.015) (1 + 0.015)3 = 1674.88 + 1650.12 + 1625.74 + 95631.70 = £100,582 . Note that in this example the interest rate 1.5% is different to the coupon rate 1.7%. 6 This picture reflects the cash-flow stream for the buyer: the buyer initially pays the issue price (negative, downwards arrow), and then receives (positive, upwards arrow) a stream of payments. The picture for the bond issuer would be the opposite: it starts with receiving the issue price, and continues with a stream of outgoing payments. 26 Recall the definition of discount factors from Section 1.4.2; these allow us to express the present value of a general bond: Example 3.5. Let D(T ) be the discount factor for a payment after T years. Then the present value of a bond that pays coupons C1 , . . . , Cn at times T1 , . . . , Tn and is redeemable at value RP at the expiry time Tn , is equal to P V = D(T1 ) · C1 + D(T2 ) · C2 + · · · + D(Tn ) · Cn + D(Tn ) · RP. 3.1.2 (14) The no-arbitrage principle and the fair price of a bond How should we assign a fair price to a bond? To answer this, modern finance takes as its starting point the assumption that there are no arbitrage opportunities in the market. This is sometimes called the no-arbitrage principle, or the ’no-free-lunch principle’. The definition of an arbitrage opportunity (or simply an ‘arbitrage’) is the following: Definition 3.6. An arbitrage is an investment that guarantees a risk-free profit (relative to discounted values). More precisely, it is the creation of a portfolio of assets and liabilities worth P (0) = 0 at time 0 such that, at some time T in the future: • The portfolio has a non-negative value P (T ) under all possible outcomes, i.e. P (T ) ≥ 0 in all possible states of the world at time T ; • There is at least one outcome under which the portfolio has a strictly positive value, i.e. P (T ) > 0 in at least one possible state of the world at time T .
Remark 3.7. Equivalently, we could describe arbitrage in terms of a portfolio worth P (0) at
time 0, and then ask whether
P (T ) ≥ P (0)D(T )−1
in all states of the world. Since P (0)D(T )−1 is the amount that a bank account with P (0)
invested in it is worth at time T , arbitrage is essentially the condition that the investment is
guaranteed to outperform the risk-free rate.
Definition 3.8. The no-arbitrage principle is the assumption, made about a financial market, that no arbitrage opportunities exist.
The justification for the no-arbitrage principle is that, in an idealised version of the market,
the buying and selling activity of market participants will eliminate all arbitrage opportunities.
This is since, the moment an arbitrage opportunity arises, there will be such demand for the
relevant assets that it will increase their prices, eliminating the arbitrage opportunity.
Note that the no-arbitrage principle may not be reflected in real-life financial markets,
especially those that are illiquid (meaning, in which trades are infrequent).
Along with the no-arbitrage principal, we will also make a series of related assumptions,
namely that participants in the market may freely create, buy, and sell assets without
restriction or cost (the so-called ‘frictionless market’ hypothesis).
It turns out that the no-arbitrage principle (combined with the ‘frictionless market’ hypothesis) is an extremely powerful tool with which to assign prices to financial instruments,
27
such as bonds; this is known as ‘arbitrage-free pricing’ or ‘rational pricing’. We will refer to
the resulting price as the ‘no-arbitrage’ price, or sometimes simply the ‘fair price’.
We begin with the the law of one price, a general expression of the no-arbitrage pricing
principle:
Theorem 3.9 (Law of one price). Under the no-arbitrage assumption, if two financial instruments generate the same cash-flow streams then they have the same current price.
In particular, the current price of a financial instrument that generates a fixed cash-flow
stream is equal to the present value of this cash-flow stream.
Corollary 3.10. The no-arbitrage price of a bond is equal to its present value.
Proof. Suppose that financial instruments A and B generate the same cash-flow streams, and
are currently priced at PA and PB respectively. Assume for the sake of contradiction that
PA > PB ;
we will show that this creates an arbitrage opportunity. Consider selling instrument A and
simultaneously buying instrument B, making a profit of
PA − PB > 0.
Since you own instrument B you will receive its associated cash-flow stream; since you sold
instrument A, you need to pay the buyer the associated cash-flow stream. As these cash-flow
streams are equal, the former can be used to pay off the latter, leaving you with a positive
profit at the expiry of the cash-flow streams. This is an arbitrage opportunity, so PA > PB
must be false. Reversing the roles of A and B, we conclude that
PA = P B .
Now suppose we have an instrument that generates a fixed cash-flow stream. We already
saw that if the present value of the cash-flow stream P V is placed in a bank account, it can be
used to exactly replicate the cash-flow stream. Hence, by the law of one price, the price of the
instrument equals the current value of the bank account, which is just the present value.
Two financial instruments that generate the same cash-flow streams are sometimes called
replicating portfolios. Indeed, the law of one price is just a formal version of the ‘replication’
arguments we saw in previous chapters.
One consequence of the law of one price is that the price of any bond can be expressed
in terms of the price of unit zero-coupon bonds (i.e. zero-coupon bonds that pay one unit
of cash at expiry). This is since, by the law of one price, the price VT of a unit zero-coupon
bond with expiry T years is equal to the discount factor to T , that is,
VT = D(T ).
Then we can use formula (14), which expresses the present value, or equivalently the noarbitrage price, of any bond in terms of the discount factors.
Example 3.11. Calculate the no-arbitrage price of a 4-year bond with face value £200,
redeemable at par, with 4% annual coupons, if the price of unit zero-coupon bonds, with
respective expiries T = 1, 2, 3, 4, are
(0.9, 0.72, 0.65, 0.53) .
28
Solution. By (14) the present value, or equivalently the no-arbitrage price, of the bond is
200 × 4% ×
4
X
D(i) + 200 × D(4),
i=1
where D(T ) is the discount factor to time T , or equivalently the price VT of a unit T -year
ZCB. Inserting the values of VT into the formula yields £128.
More generally, the no-arbitrage principle can also be used to fix relationships between
different financial quantities that are not strictly speaking financial instruments, such as interest
rates. We illustrate this with an example:
Example 3.12. Suppose that bank A offers us a monthly compounded interest rate r1M ,
and bank B offers a yearly compounded interest rate r1Y . Show that that the no-arbitrage
principle requires that
12

1
1 + r1Y = 1 + r1M .
12
First Solution. Assume for the sake of contradiction that
12

1
1 + r1Y < 1 + r1M , 12 (15) (the reverse inequality is treated similarly). Then suppose we took out a loan of £1 in bank B, which would require us to repay (1 + r1Y ) in one year’s time. We could then invest this money 12 1 in bank A. At the end of the year the account value would grow to 1 + 12 r1M , which after paying back the loan gives us a profit of 12 1 − 1 + r1Y . 1 + r1M 12 Since by assumption this is strictly positive, we have made a risk-free profit which contradicts the no-arbitrage principle. Therefore inequality (15) cannot hold. Second Solution. Put 1 1 1+ 12 r1M 12 in bank A and 1 1+r1Y in bank B. Both invest investments result in a unit pay off after 1 year. By The Law of One Price, 1 or 1+ 1 r 12 1M 12 = 1 , 1 + r1Y 12 1 1 + r1Y = 1 + r1M . 12 Exercise 3.13. Explain why, in the example above, the reverse inequality 12 1 1 + r1Y > 1 + r1M
12
29
3.1.3
The bond yield
If the interest rate is unknown, we cannot compute the present value. However, if we know
the issue price of a bond, we can use no-arbitrage theory to deduce the ‘hypothetical’ interest
rate that would make the bond fairly price. This is called the bond yield, and is nothing more
than the ‘internal rate of return’ of an investment in the bond.
Definition 3.14. Suppose that a bond has issue price P0 , coupon payments of C1 , . . . , Cn at
times t1 , . . . , tn and redemption payment RP at expiry T . Then the bond yield is the unique
solution r ∈ (−1, ∞) to
n
X
1
Ci
+
RP.
(16)
P0 =
t
i
(1 + r)
(1 + r)T
i=1
Equation (16) has a unique solution for the same reason that the IRR is guaranteed to
exist (see Theorem 1.25). Just like the IRR, the bond yield can be used to determine whether
the bond is a good investment.
Example 3.15. A broker proposes to sell a 2-year bond with face value £100,000, redeemable
at par, with annual coupon payments of 2%. The broker quotes an issue price of P0 =
£102,000. Find the yield of the bond, and decide whether the bond is a good investment if
the interest rate is 10%.
Solution. The yield is the (unique) solution r ∈ (−1, ∞) to the equation
102,000 =
2,000
100,000
2,000
+
+
.
2
1 + r (1 + r)
(1 + r)2
This is a quadratic in the variable x = 1 + r; solving this yields
x = −0.9902
or
1.0098.
Since x > 0 we can disregard the first solution, and we are left with
r = x − 1 = 0.98%.
Since this is far less than the interest rate, the bond is not a good investment.
As shown by the example, the yield does not equal the coupon rate in general.
3.2
The term structure of interest rates
So far it has been assumed that there is a single interest rate r = r(t) at each instant of
time, regardless of the type or duration of the deposit/loan. In practice, interest rates usually
depend on the term (i.e. the duration) of the deposit/loan. The main reason for this is supply
and demand: if many people are asking for 20-year loans, then the the interest rate for 20-year
deposits will increase. Likewise, if there is a good supply of one month deposits the 1-month
interest rate will go down. This variation is referred to the term structure of interest rates.
For example, the sterling LIBOR rates7 on 17th September 2019 were:8
7
LIBOR stands for London Interbank Offered Rate, and is a set of benchmark interest rates fixed every day
at 11am London time. LIBOR is used by many institutions to set short-term interest rates; the total value of
instruments fixed by LIBOR is estimated to be in the hundreds of trillions of pounds.
8
Source: The Intercontinental Exchange Benchmark Administration Ltd https://www.theice.com/
marketdata/reports/170. In this page you can consult current or past LIBOR rates for a number of
currencies, e.g. GBP, Euros, Yen and USD.
30
Term
Overnight
1 Week
1 Month
2 Month
3 Month
6 Month
1 Year
LIBOR
0.67150%
0.68825%
0.71225%
0.75775%
0.78463%
0.84050%
0.96213%
The fact that interest rates are term-dependent is an additional complication when pricing
fixed-rate financial instruments. In this section we explore how these rates can be analysed.
In particular, we use the price of unit zero-coupon bonds to relate various interest rates,
via the no-arbitrage principle.
3.2.1
Spot rates
The first step in analysing term-dependent interest rates is to convert them to effective
rates, analogous to the ‘effective rates’ we defined in Section 1.1 for different compounding
frequencies. The name for these effective rates, in the context of term-dependent interest
rates, is spot rates.
By default, interest rates are quoted as (annualised) nominal rates; this is the case for
the LIBOR rates quoted above. This means that if you place P in a deposit for term T , at
interest rate rT , you will get back
P (1 + rT T ).
A fairer way to compare term-dependent rates is to compare their spot rates:
Definition 3.16. The spot rate, sT , for maturity T is defined as
sT = (1 + rT T )1/T − 1
(17)
or, equivalently,
(1 + sT )T = 1 + rT T.
The function sT is called the spot rate curve.
Remarks 3.17. Equation (17) can be compared to (3), which was the formula that we used
to define ‘effective rates’ for various compounding frequencies. Just as for effective rates, spot
rates are also ‘artificial’, in the sense that they are not directly quoted on the market.
Examples 3.18.
1. The 1-year spot rate s1 is equal the nominal rate r1 .
2. The 2-year spot rate s2 is equal to rate such that, compounding twice, has the same
effect as the nominal 2-year rate r2 , i.e.
(1 + s2 )2 = 1 + r2 · 2.
3. The 6-month spot rate s0.5 has the same effect as the 6-month nominal rate compounded
twice, i.e.

1
1 2
⇔ (1 + s0.5 )1/2 = 1 + r0.5 · .
1 + s0.5 = 1 + r0.5
2
2
31
Example 3.19. Let r1/12 = 0.712% be the 1-month LIBOR rate quoted above. Then the
associated spot rate satisfies

1 12
1 + s1/12 = 1 + 0.712%
= 1.00714
12
which yields a spot rate of sT = 0.714%.
In the case of term-dependent interest rates, the discount factors D(T ) that we use to do
present value analysis must be computed in terms of the nominal/spot rates
D(T ) = (1 + rT T )−1 = (1 + sT )−T .
Example 3.20. Calculate the no-arbitrage price of a five-year bond with face value £100,000,
redeemable at par, with 6% annual coupons, if the spot rate curve is
(s1 , s2 , s3 , s4 , s5 ) = (7%, 7.25%, 7.5%, 7.75%, 8%) .
Solution. We use the discount factor D(i) = (1 + si )−i in the formula (14). This yields a
no-arbitrage price of
6,000
5
X
D(i) + 100,000D(5) = £92,247.
i=1
Another point of view is that, under the no-arbitrage assumption, spot rates are determined
by the price of unit zero-coupon bonds VT with expiry T . This is since
−1/T
sT = D(T )−1/T − 1 = VT
− 1.
(18)
Examples 3.21. The following table gives some prices for unit zero-coupon bonds with varying
maturities, alongside the implied spot rate calculated via the formula (18):
T =Year
1
5
10
15
Price
0.94
0.70
0.47
0.30
Spot rate sn
6.4%
7.4%
7.8%
8.4%
Remark 3.22. Observe that the spot rates sT we defined play the same role as the effective
rate reff , in the sense that they define an equivalent annualised interest rate. Some practitioners
instead convert rT to an instantaneous spot rate s0T that is the equivalent continuouslycompounded rate. This satisfies
0
esT T = 1 + rT T
3.2.2
s0T =

1
ln (1 + rT T ) .
T
Forward rates
How much interest should we expect to earn on money deposited or loaned between two
dates in the future? Surprisingly, the fair (i.e. no-arbitrage) value of such a ‘forward rate’ is
completely determined by current spot rates.
Why is this? Suppose I expect to have £1m in one year’s time and that I wish to deposit it
for a further year. How much interest might I expect to receive? Well consider instead taking
the following sequence of actions:
32
t=0
• Take out a 1-year loan for (1+s1 )−1 × £1m. Place this amount in a 2-year deposit.
t=1
• Pay £1m that you expect to have at this time to pay off the loan. Think of this
money as an investment that will receive a return at time t = 2.
t=2
• The deposit will now be worth (1 + s2 )2 (1 + s1 )−1 ×£1m.
Under this procedure, the £1m we expect to have in one year’s time will grow to £1m
to (1 + s2 )2 /(1 + s1 )×£1m at the end of the second year. This implies that the money has
earned an ‘effective’ interest rate f1,2 satisfying
1 + f1,2 =
(1 + s2 )2
.
1 + s1
Let us now formalise this using no-arbitrage principles.
Definition 3.23. A forward rate agreement (FRA) from time n1 to time n2 is a contract
whereby a party agrees to place a certain amount (the principal) in a deposit or a loan at time
n1 for an effective interest rate of f , and the deposit/loan expires at time n2 . In particular, a
deposit of P at time n1 will be worth, at the maturity time n2 ,
P (1 + f )n2 −n1 .
A signatory to a FRA is obliged to deposit (or loan) the money at the start time n1 (i.e. there
is no optionality).
Intuitively the fair value of f in a FRA might be thought of as both parties best guess
as to what the interest rate will be at time n1 . However this is incorrect, since as we argued
before f is completely determined by today’s spot rates.
Proposition 3.24. The fair (i.e. no-arbitrage) interest rate for a FRA from time n1 years to
n2 years is
1
(1 + s )n2 n −n
n2
2
1
fn1 ,n2 =
− 1,
n
1
(1 + sn1 )
where sn1 and sn2 are the spot rates to n1 and n2 years. Equivalently,
(1 + fn1 ,n2 )n2 −n1 =
(1 + sn2 )n2
.
(1 + sn1 )n1
Proof. The proof consists of formalising the previous argument using the no-arbitrage principle.
Suppose for the purposes of contradiction that a bank proposes a FRA with forward rate f
that is larger than the forward rate fn1 ,n2 . Then at time 0 we can (i) sign this FRA for a
deposit between times n1 and n2 , (ii) simultaneously take out a loan with term n2 of amount
(1 + sn1 )−n1 ,
and (iii) deposit this amount immediately into a bank account for term n1 . At time n1 , the
deposit will have grown to amount
(1 + sn1 )−n1 × (1 + sn1 )n1 = 1.
At this point we withdraw this money from the bank, and use it to fulfil the FRA. At time n2 ,
this will now be worth
(1 + f )n2 −n1 > (1 + fn1 ,n2 )n2 −n1 =
33
(1 + sn2 )n2
,
(1 + sn1 )n1
where the first inequality is by the assumption, and the second is the definition of the forward
rate fn1 ,n2 . At the same time, our initial loan has grown to value
(1 + sn2 )n2
(1 + sn1 )n1
so we can use the deposit to pay off the loan, generating a risk-free profit. A similar argument
2σ 2
2πσy
fY (y) =
0,
if y ≤ 0.
Let us consider the growth of a single deposit in the case that 1 + Ri are assumed to be
an i.i.d. sequence of log-normally distributed random variables. In that case the distribution
of
Sn = (1 + R0 )(1 + R1 ) · · · (1 + Rn−1 )
can be computed exactly (and not merely the mean and variance as we have previously done).
The underlying reason is that log-normal distributions behave well under multiplication. Suppose that Y1 ∼ LogNormal(µ1 , σ12 ) and Y2 ∼ LogNormal(µ2 , σ22 ) are independent. Then
we can write Y1 = eX1 and Y2 = eX2 , where X1 ∼ N (µ1 , σ12 ) and X2 ∼ N (µ2 , σ22 ) are
independent. Therefore,
Y1 Y2 = eX1 eX2 = eX1 +X2 ,
and since
X1 + X2 ∼ N (µ1 + µ2 , σ12 + σ22 ),
it follows that
Y1 Y2 ∼ LogNormal(µ1 + µ2 , σ12 + σ22 ).
By extending the previous results to n variables we can show the following:
40
Theorem 4.8. Suppose that 1 + Ri are a sequence of i.i.d. random variables with 1 + Ri ∼
LogNormal(µ, σ 2 ). Then
Sn =
n−1
Y
(1 + Ri ) ∼ LogNormal(nµ, nσ 2 ).
i=0
Using the above theorem, the c.d.f. of Sn can be computed as
X − nµ
ln x − nµ
X
P(Sn ≤ x) = P(e ≤ x) = P(X ≤ ln x) = P √
≤ √

where X ∼ N (nµ, nσ 2 ). Therefore
P(Sn ≤ x) = Φ
ln x − nµ

,

where Φ denotes the c.d.f. of the standard normal distribution.
Example 4.9. Suppose £1000 is deposited in a bank account for five years, and suppose
the interest rates Ri are a sequence of i.i.d. random variable such that 1 + Ri is log-normally
distributed with parameters µ = 0.075 and σ 2 = 0.0252 . What is the probability that the
accumulated value of the deposit exceeds £1,500? Find the upper and lower quartiles for the
accumulated value of the deposit.
Solution. The accumulated value An = 1000S5 has distribution
An ∼ 1000LogNormal(5µ, 5σ 2 ).
The probability that An > 1500 is
ln 1.5 − 5µ

P(An > 1500) = P(LogNormal(5µ, 5σ 2 ) > 1.5) = 1 − Φ

= 1 − Φ(0.545) = 1 − 0.7017 = 0.2929,
where the value of the normal c.d.f. has to be looked up (in a table for instance).
By definition of the quartile, the accumulated value An will exceed the upper quartile u
with probability 0.25, i.e.
0.75 = P(An ≤ u) = P(1000S5 ≤ u) = P(S5 ≤ u/1000).
Hence u satisfies

ln(u/1000) − 5µ

0.75 = Φ
.

Looking up the normal c.d.f. we find that Φ(0.6745) = 0.75, so therefore
ln(u/1000) − 5µ

= 0.6745,

that is,

u = 1000 exp 5µ + 0.6745σ 5 = £1,510 .
Similarly, the lower quartile is

` = 1000 exp 5µ − 0.6745σ 5 = £1,400 .
41
5
Equities and their derivatives
In previous chapters we considered fixed-income securities. Equities and derivatives differ
from fixed-income securities in that their cash-flow stream is not pre-determined, and instead
depends on market fluctuations. This complicates their analysis.
A derivative is a financial instrument (i.e. a contract between two or more parties) whose
value is ‘derived’ from one or more underlying assets (e.g. stocks, bonds, foreign exchange,
interest rates, commodities etc). There exist many types of derivatives; in this course we
study only the most common examples being forwards and options, but there are many
other ‘exotic’ derivatives. Derivatives can be bought and traded either through an exchange
such as the Chicago Mercantile Exchange (CME) or over-the-counter (off-exchange). The
most common use of derivatives is to reduce expose to risk (‘hedge’), or conversely to increase
exposure to risk (‘speculate’).
5.1
Shares
The basic instrument of equity is the share (or ‘stock’),9 which is a unit of ownership of a
company. Owning shares translates to partly owning the company which issues the shares.
Example 5.1. Apple currently has 4.52 billion shares. If you were to buy one share in Apple
(current price \$240.24) you would own
1
= 0.0000000221%
4,520,000,000
of the company. The current market capitalisation of Apple (i.e. the total value its shares) is
4.52 billion × \$240.24 = \$1,086 trillion.
Sometimes shares entitle the holder to additional benefits, for instance to dividends (cash
payments made by the company to its shareholders). For simplicity, we will generally only
consider non-dividend paying shares.
Shares are usually traded on a stock exchange, such as the London Stock Exchange or
the New York Stock Exchange. We will always assume that:
1. Shares can be bought and sold by an investor in any amount (even fractional amounts),
at any time, without transaction costs.
2. An investor can borrow a share and then subsequently sell it. This operation is called
short-selling, and can be viewed as owning a negative number of shares.
These assumptions are another instance of the ‘frictionless market hypothesis’.
5.2
Forwards
A forward (or a forward contract) is an agreement between two parties in which the seller
promises to sell a specified asset (the ‘underlying’) for a specified price F (the ‘forward price’)
at a specified time T (the ‘settlement date’) in the future, while the buyer promises to buy it.
Usually the underlying is either a commodity (e.g. oil, gas, gold etc.) or foreign exchange.
9
These terms are basically synonymous; shares is mainly used in the UK, while stock or common stock is
mainly used in the US.
42
Forward contract are not bought or sold, but are ‘entered into’. The investor can either
enter a long position in the forward, if they are buying the underlying asset at time T , or a
short position in the forward, if they are selling the underlying asset at time T .
As with all derivatives, forward contracts can be used to:
• Reduce exposure to risk, i.e. to hedge. Suppose that a company needs 1,000 barrels
of oil in 6 months’ time, and wants to eliminate the risk that the oil price will move
significantly higher by then. Or suppose they trade in the UK but need to buy oil in US
dollars. By entering into a forward contract they can fix the price of the future purchase,
and so eliminate their exposure to price/currency fluctuations.
• Increase exposure to risk, i.e. to speculate. Suppose that an investment bank expects
the price of oil to increase, but has no need for oil. By entering into a forward contract,
they can agree to buy oil at a fixed (low) price, and then at the settlement date, when
the oil is received, they can immediately sell for a profit (in practice, they would never
even take possession of the oil).
Example 5.2. Suppose a company enters into a forward contract with an oil supplier to
purchase 1,000 barrels of oil on 1st January 2020 for \$42,000. Suppose that the price of oil
per barrel on 1 January 2020 is \$50. Then the company, who agreed to buy 1,000 barrels on
1 January for \$42,000, could immediately sell the oil for \$50,000, yielding an instant profit of
1,000 · \$50 − \$42,000 = \$8,000.
Conversely, if the price on 1 January 2020 is \$35, then on 1 January the company takes an
instant loss of
1,000 · \$35 − \$42,000 = −\$7,000.
Even though two companies are free to enter into a forward contract at any forward price F
they agree on, in reality the principles of no-arbitrage pricing determine a single ‘fair’ forward
price for every asset:
Proposition 5.3. Suppose that an asset has current price S at time t = 0. By the noarbitrage principle, the forward price F agreed at t = 0 for the delivery of the underlying asset
at t = T must be equal to
F = S/D(T )
where D(T ) is the discount factor to time T .
Proof. We prove this result using ‘replicating portfolios’ (i.e. we apply the law of one price).
Consider the following two portfolios formed at t = 0:
• Portfolio A. A portfolio consisting of the asset.
• Portfolio B. A portfolio consisting of:
– a long position in the forward contract; and
– T -year zero-coupon bonds with face-value F (or, equivalently, D(T )F deposited
in a bank account).
43
We claim that both portfolios will have the same value at time T . To see this, observe that the
value of Portfolio A is equal to the value of the asset at time T . On the other hand, Portfolio
B also consists of the asset (guaranteed through the forward contract), less the forward price
F that needs to be paid, plus the F that the bond pays upon maturity. It total, we see that
Portfolio B has the same value as the asset at time T .
Since portfolios A and B are replicating, by the law of one price they have the same price
at t = 0. Let us now examine these prices. The price of Portfolio A at time 0 is equal to the
price of the asset, which is S. The price of Portfolio B is equal to the price of the zero-coupon
bond (remember that we do not pay anything to enter into the forward contract), which is
D(T )F . Equating these, we conclude that S = D(T ) F ⇔ F = S/D(T ).
The current price of an asset of often referred to as the spot price. It is equal to the
forward price of a contract with immediate delivery.
Example 5.4. Interest is continuously-compounded at nominal rate r = 5%. What is the fair
forward price F for a contract to buy 1,000 barrels of oil in one year’s time if the spot price
is \$37 per barrel.
Solution. Since D(1) = e−0.05 = 0.9512, the fair forward price is
F = 37, 000/0.9512 = \$38, 897 .
Example 5.5 (Not covered in lectures). Suppose that the effective rate of interest in the UK
is r£ = 0.5%, and in the Eurozone is r€ = 1%. Suppose also that the current exchange rate
is €1.13 to £1. What is the fair forward price £F to buy €1,000 in three months’ time?
Solution. By the above proposition,
F = S/D(1/4),
where S is the current price (in pounds) of € 1,000
S=
1000
= 885,
1.13
determining the relevant discount factor D(T ), since there are two interest rates: r£ and r€ .
To resolve this, let us take a more direct approach using ‘replicating portfolios’. Consider
two portfolios, both of which will be worth €1,000 at time T = 1/4:
• Portfolio (A). A long position in a forward contract to buy € 1,000 at time T , and
£ F/(1 + r£ )T deposited in a UK bank until time T .
• Portfolio (B). € 1,000/(1 + r€ )T deposited in a bank in Germany until time T .
By the law of one price, since the two investments are worth the same amount at time T , they
are also worth the same amount at time 0. Thus, in pounds,
0 + F/(1 + r£ )T =
1,000/(1 + r€ )T
1.13
which implies
1,000 1 + r£ T
1,000 1.005 1/4
F =
=
= £883.9.
1.13 1 + r€
1.13 1.01
44
Here we see that the relevant discount factor was
1 + r −T
£
D(T ) =
,
1 + r€
which implies that value is ‘discounted’ with respect to the ‘currency adjusted’ rate
1 + r£
r£ − r€
−1=
= −0.495%.
1 + r€
1 + r€
(21)
Note that the expression (21) is mathematically the same as the ‘inflation adjusted’ interest
rate we saw in Section 1.3, with r€ playing the role of the inflation rate.
5.3
Options
An option is a derivative contract that gives the buyer the right, but not the obligation, to
buy or sell an underlying asset for a specific strike price K (also known as the ‘exercise price’)
on, and sometimes also prior to, the expiry time T (also known as the ‘maturity’). The seller
of the option is then obliged to fulfil the transaction. The buyer pays a certain fee to purchase
an option.
Options are traded at the Chicago Board Options Exchange (CBOE), or over-the-counter
as customised contracts between a single buyer and seller.
Remark 5.6. Be sure you understand the difference between a forward contract and an option.
A forward contract gives you the obligation to buy or sell an underlying asset for a specified
price (the forward price F ), while an option gives you the right, but not the obligation, to
buy or sell an underlying asset for a specified price (the strike price K). On the other hand,
forward contracts are free to enter, whereas there is a cost to buy an option.
There are two main categories of options:
• Call options, which give the owner the right to buy the asset at the strike price;
• Put options, which give the owner the right to sell the asset at the strike price.
These are further divided into two main categories:
• European options which give the owner the right to buy or sell the asset at the strike
price at the expiry date.
• American options, which give the owner the right to buy or sell the underlying asset
at the strike price at any time up to and including the expiry.
Example 5.7. A European call option on a share, with strike K and expiry T , is a contract
that gives the right, but not the obligation, to purchase the share at time T for price K. A
European put option is similar, but the right to purchase is replaced by a right to sell.
Example 5.8. An American call option on a share, with strike K and expiry T , is a contract
that gives the right, but not the obligation, to purchase the share at any time up to and
including time T for price K. An American put option is similar, but the right to purchase is
replaced by a right to sell.
American options are more expensive (or at least as expensive) than the equivalent European options (i.e., with the same expiry T and strike K), since they offer their holder more
optionality.
45
Exercise 5.9. Use a no-arbitrage argument to verify that an American option must cost at
least as much as the equivalent European option.
Remarks 5.10.
1. The options described above are actually only a small sample of the enormous variety
of options that are issued and traded. For this reason they are sometimes referred to as
‘vanilla’ options, with other variety of options known as ‘exotic’.
2. Although options can be written on any underlying asset, for simplicity we will work only
with options written on an underlying share.
3. In practice an option will …

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