TH6154: Financial Mathematics 1
Term 1, 2022-23
Contents
Preface 2
1 Interest rates and present value analysis 4
1.1 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Variable interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The instantaneous interest rate . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 The yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Present value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Defining the present value . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.3 Balancing present values . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Rates of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 The annualised rate of return and the equivalent effective interest rate 14
1.5.2 Internal rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Immunisation of assets and liabilities 18
2.1 Measuring the effect of varying interest rates . . . . . . . . . . . . . . . . . . 18
2.2 Reddington immunisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Immunisation in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Immunisation when interest is compounded yearly . . . . . . . . . . . . . . . 23
3 Bonds and the term structure of interest rates 25
3.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 The present value of a bond . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 The no-arbitrage principle and the fair price of a bond . . . . . . . . . 27
3.1.3 The bond yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 The term structure of interest rates . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Spot rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Stochastic interest rates 36
4.1 A fixed interest rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 A varying interest rate model . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 The growth of a single deposit . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 The growth of regular deposits . . . . . . . . . . . . . . . . . . . . . 39
4.3 Log-normally distributed interest rates . . . . . . . . . . . . . . . . . . . . . . 40
1
5 Equities and their derivatives 42
5.1 Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Pricing options: model-independent pricing principles . . . . . . . . . . . . . . 47
5.5 Pricing options via the risk-neutral distribution . . . . . . . . . . . . . . . . . 49
5.5.1 Betting strategies, the risk-neutral distribution, and the arbitrage theorem 49
5.5.2 Options pricing using the risk-neutral distribution . . . . . . . . . . . . 53
6 The binomial model 55
6.1 The single-period binomial model . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.1 The no-arbitrage condition and the risk-neutral distribution . . . . . . 55
6.1.2 Option pricing in the single-period binomial model . . . . . . . . . . . 57
6.1.3 Replicating portfolios and the delta-hedging formula . . . . . . . . . . 60
6.2 The two-period binomial model . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.1 Options pricing in the two-period binomial model . . . . . . . . . . . . 62
6.2.2 Replicating portfolios in the two-period binomial model . . . . . . . . 68
6.2.3 Option pricing via the two-period risk-neutral distribution . . . . . . . 68
6.3 The multi-period binomial model . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3.1 Options pricing in the multi-period binomial model . . . . . . . . . . . 70
7 From the multi-period binomial model to the Black-Scholes formula 73
7.1 General discrete-time models . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2 The log-normal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3 The Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.3.1 Properties of the Black-Scholes formula . . . . . . . . . . . . . . . . . 77
7.4 The log-normal process as an approximation of the bin. model . . . . . . . . . 78
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 Man-
agement1) 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:
1Formally 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
trade-off between accuracy and simplicity.
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 princi-
ple. These concepts aremodel-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
1
n
-th of a year. This means that every 1
n
-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. (2)
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).
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
PWe can generalise the notion of present value to a cash-flow stream a = (a1, a2, . . . , an)
that pays ai at the end of year i, for i = 1, . . . , n. The present value.
To see this, observe that the cash-flow stream a = (a1, a2, . . . , an) can be replicated by
first splitting the stream into the individual payments a1, a2, . . ., and then depositing the
corresponding amounts PV (a1), PV (a2), . . . needed to replicate these payments. Since
PV (a) = PV (a1) + PV (a2) + . . .+ PV (an),
the total amount you need to deposit to replicate the cash-flow stream is PV (a).
Remark 1.13. This is our first example of an argument that uses the idea of replication
to assign a value to a financial instrument. Later in the course we will formalise this type of
argument by using the no-arbitrage assumption.
Example 1.14. You are offered three different jobs. The salary paid at the end of each year
(in thousands of pounds) is
Job\Year 1 2 3 4 5
A 32 34 36 38 40
B 36 36 35 35 35
C 40 36 34 32 30
Which job pays the best if the interest rate is (i) r = 10%, (ii) r = 20%, or (iii) r = 30%?
4The term net present value (NPV) is sometimes used when discussing cash-flow streams as opposed to
a single payment, but we will not use this term.
10
Solution. We shall compare the present values of the cash-flow streams. The present value
for job A is
We can generalise the notion of present value to a cash-flow stream a = (a1, a2, . . . , an)
that pays ai at the end of year i, for i = 1, . . . , n. The present value
4 of a is
PV (a) =
n∑
i=1
ai
(1 + r)i
.
To see this, observe that the cash-flow stream a = (a1, a2, . . . , an) can be replicated by
first splitting the stream into the individual payments a1, a2, . . ., and then depositing the
corresponding amounts PV (a1), PV (a2), . . . needed to replicate these payments. Since
PV (a) = PV (a1) + PV (a2) + . . .+ PV (an),
the total amount you need to deposit to replicate the cash-flow stream is PV (a).
Remark 1.13. This is our first example of an argument that uses the idea of replication
to assign a value to a financial instrument. Later in the course we will formalise this type of
argument by using the no-arbitrage assumption.
Example 1.14. You are offered three different jobs. The salary paid at the end of each year
(in thousands of pounds) is
Job\Year 1 2 3 4 5
A 32 34 36 38 40
B 36 36 35 35 35
C 40 36 34 32 30
Which job pays the best if the interest rate is (i) r = 10%, (ii) r = 20%, or (iii) r = 30%?
4The term net present value (NPV) is sometimes used when discussing cash-flow streams as opposed to
a single payment, but we will not use this term.
10
Solution. We shall compare the present values of the cash-flow streams. The present value
for job A is