MTH6154: Financial Mathematics 1
Term 1, 2022-23
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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
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
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.
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 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.
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
-th of a year. This means that every 1
-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
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
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
2A security is a tradable financial instrument.
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
= 100× 1.195 = £119.5.
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
(amount after one year)− P
For an interest rate compounded every 1
-th of a year, we therefore have
P (1 + r/n)n − P
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
119.5− 100
= 0.195 = 19.5%.
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
− 1 = 0.2775.
• For company B, the effective rate is:
− 1 = 0.2770.
3Effective rates are sometimes also called ‘annual percentage rates’ (APRs).
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
-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
(we prove this rigorously in Coursework 1). The effective rate for continuously-compounded
interest at nominal rate r is therefore
Example 1.5. Suppose that a bank offers continuously-compounded interest at nominal
rate 5%. The effective rate is
0.05 − 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)
For example, if interest is compounded n times a year, this is
and if it is compounded continuously
P (er)T = PerT .
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
PerT = 2P ⇒ erT = 2 ⇒ rT = ln 2,
and therefore
For example, if the nominal rate is r = 12%, it takes 5.78 years for the amount to double.
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.
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 instanta-
neous interest rate. We always assume that the instantaneous interest is continuously-
compounded, 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
er(t)h = 1 + r(t)h+
r(t)2h2 + . . . ≈ 1 + r(t)h.
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
Theorem 1.8. Suppose that r(t) is a piece-wise continuous function. Then
P ′(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).
As h becomes smaller, the above approximation becomes more and more accurate, and so
P (t+ h)− P (t)
= P (t)r(t).
Recognising the left-hand side as the definition of the derivative, we conclude that
P ′(t) = P (t)r(t),
The above equation is a separable first order differential equation, easily solved by writing
and integrating both sides: ∫ t
r(u) du =⇒ lnP (t) = lnP (0) +
P (t) = P (0) exp
which is the desired relation between the amount P (t) in your bank at time t and the time-
varying 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.
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):
The yield curve allows us to write the formula for P (t) in (4) as
P (t) = P (0) exp
= P (0)er̄(t)t.
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 de-
posited/loaned, and is equal to
reff(T ) = e
r̄(T ) − 1.
Example 1.9. Suppose you deposit £100 in a bank offering the instantaneous interest rate
Find the yield curve r̄(t) and determine the value of the bank balance at time t = 3.
Proof. To calculate the yield curve we split the integral at t = 2. For t < 2, 3% ds = 3% whereas for t ≥ 2, 0.05(t− 2) The value of the deposit is 100er̄(3)3 = 100e(0.05−0.04/3)3 = 100e0.11 = £112. Example 1.10. Suppose a bank offers a loan with instantaneous interest rate increasing slowly from 3% to 4% via the function Find the effective rate for a loan until time T = 5. Proof. We can rewrite the interest rate as 4%× (1 + t) The yield curve is 4%× t− 1%× ln(1 + t) ln(1 + t). The effective rate for the loan is reff(5) = e r̄(5) − 1 = e0.04− ln(6) − 1 = 3.71%; as expected, this lies somewhere between 3% and 4%. 1.3 Inflation We finish the chapter by taking a quick look at inflation, the phenomenon of prices increasing as a whole over time (or equivalently, the purchasing power of money eroding over time). Inflation is most easily quantified by reference to an index, for example the Retail Price Index (RPI), which tracks the price of a basket of goods. Let rinf be the (annualised) inflation rate, which means that £1 today will have the purchasing power of £(1 + rinf) in one year. For example, if rinf = 3%, then a basket of goods costing £100 today, will cost £103 next year. Equivalently, the purchasing power of £103 in one year is equivalent to £100 today. How should we take inflation into account when assessing interest rates? Suppose you deposit the amount P today. After one year the deposit will have grown to P (1+ r), where r is the interest rate. On the other hand, the purchasing power of the amount P (1 + r) in one year from now, taking into account inflation, is the same as the amount P (1 + r) (1 + rinf) today. So the ‘real’ interest rate (real in the sense that it reflects the decrease of purchasing power), also known as the inflation adjusted interest rate, is equal to Example 1.11. Suppose that the interest rate is 5% and the rate of inflation is 3%. Then the inflation adjusted interest rate is 0.05− 0.03 = 0.0194 = 1.94% . In the case of compounding interest, we must use the effective interest rate in place of the nominal rate, i.e., reff − rinf Example 1.12. Suppose that the interest rate is 5% compounded monthly, and the rate of inflation is 3%. Then the inflation adjusted interest rate is reff − rinf ((1 + 0.05 )12 − 1)− 0.03 = 0.0205 = 2.05%. Since rinf is usually much smaller than 100%, it is often sufficient to use the approximation reff − rinf ≈ reff − rinf. 1.4 Present value analysis A cash-flow stream is a sequence of payments that are specified by their amount and transaction time. Often, but not always, the cash-flow stream will consist of payments at regular intervals (for instance, one per year), in which case we can write the cash-flow stream simply as a = (a1, a2, . . . , an), where ai denotes the amount paid at the end of period i. The question we consider in this section is: How should we compare different cash-flow streams? For this we introduce the concept of the present value of a cash-flow stream. 1.4.1 Defining the present value Suppose that the interest rate is 10%. If somebody gives you £100 today, you could put it in the bank and after a year you would have £110. Thus £100 received today has a value of £110 in a year’s time, or conversely, £110 received in a year’s time has a present value of More generally, suppose that the inter 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com