MATH580 The Vasicek model for interest rates
The Vasicek model for interest rates
MATH580 PROJECT
1. The Black-Scholes model for stock prices
The Black-Scholes model describes the price of stock at time t as a stochastic process
{St : t ≥ 0} determined by the Geometric Brownian Motion (GBM).
A European call option on the stock with strike price c and expiry time t0 is a con-
tract that gives the buyer the right (but not the obligation) to buy one unit of the stock
at time t0 at price c. The buyer will only exercise this option if St0 > c, in which case the
payoff is given by C = (St0 − c)+ = max{St0 − c, 0}. The classical Black-Scholes formula
establishes the price at time 0 of this European call option as
Pt0 = E[e
−ρt0C] = E[e−ρt0(St0 − c)+]
where ρ is the interest rate, and under the assumption that µ = ρ− σ2/2. Note that this
choice of µ is the one that means the expected discounted stock price is constant.
This formula can be written as
Pt0 = S0Φ
(
log(S0/c) + (ρ+ σ
2/2)t0
σ
√
t0
)
− ce−ρt0Φ
(
log(S0/c) + (ρ− σ2/2)t0
σ
√
t0
)
Question 1-1. Plot {Pt : 0 ≤ t ≤ 10} against time for S0 = 1, σ = 0.02, ρ = 0.03 and
c = 1 and comment on this.
Question 1-2. Investigate how the price Pt0 varies with σ, c and ρ. For example,
plot Pt for t = 10 as you vary each of σ, c, and ρ in turn. Comment on the results you
get, including why Pt behaves as it does.
NB. You do not need to do any simulation for this part of the project. You may use
any programming language as you wish.
2. The Vasicek model for interest rates
In the Black-Scholes model, the assumption was made that the interest rate was fixed
and risk-free. In practice this is not the case and several models exist that attempt to
describe interest rates mathematically.
The Vasicek model for the interest rate rt states that:
drt = (α− βrt)dt+ σdWt,
where Wt is a standard Brownian Motion.
Question 2-1. Solve the following (without using a computer/calculator)
(i). Apply Itoˆ’s lemma to derive d(eβtrt).
(ii). Using (i) or otherwise, find the expression for rt. State the expected value of rt,
derive its limit as t→∞, and discuss why this model explains the important mean
reverting characteristic of interest rates.
Question 2-2. Simulate {rt : 0 ≥ t ≥ 10} for r0 = 0.1, β = 0.5, α/β = 0.05 and σ = 0.02,
and plot realisations of this. Plot E(rt) and V ar(rs) against time. What do you notice?
Can you explain this? Is this in line with what you find in Question 2-1? What happens
if you change the values of the parameters?
3. The report
This project contributes 20% to the overall assessment for MATH580. Each project
should consist of a written report which should be submitted via Moodle, as a pdf file,
by the deadline specified on the Moodle Page. The written report should contain all
computing code used as an appendix. Reports which contain computer code in the main
body will be penalised. The report should be no more than 3 pages (excluding the ap-
pendix), written in size 12 font with sensible margins and spacing. This includes all
tables, graphs; but excludes the appendix. Reports over 3 pages will be penalised.
A shorter report, with material effectively communicated, is acceptable. Marks will be
assigned for the presentation, organisation and content of your report. Your goal is to
carry out the investigations of the financial stochastic processes outlined above. For the
model, in addition to the graphs and explanations of your findings, you should provide
some background information and a brief discussion of benefits and shortcomings of the
models. Remember to reference this appropriately.
2
4. Selected References
Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities”.
Journal of Political Economy 81 (3): 637-654.
Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ:
Prentice Hall. ISBN 0-13-009056-5.
Vasicek, Oldrich (1977). “An Equilibrium Characterisation of the Term Structure”. Jour-
nal of Financial Economics 5: 177-188.