2MTH5210 PRACTICE Exam.
You can use 1 double-sided A4 sheet of hand written notes.
There are five questions, 20 marks each.
In the following, Bt, t ≥ 0, denotes the standard Brownian motion process started
at zero.
1. Let Vt = e
∫ t
0
µ(s)ds+
∫ t
0
σ(s)dBs , where µ(t) and σ(t), 0 ≤ t ≤ T , are deterministic
continuous functions.
(a) Show that Xt = log Vt is a Gaussian process with independent increments.
(b) Find the quadratic variation of Vt. [5 marks]
(c) Give the necessary and sufficient condition on the functions µ(t) and σ(t),
so that the process Vt is a martingale. Justify your answer. [5 marks]
(d) It is given that for any u ∈ R the process eu log Vt−u24 t3 is a martingale. Find
µ(t) and σ2(t). [5 marks]
2. Consider the following market model. Let St = Bt+ t
2 be the price process of
an asset and let βt = 1, be a savings account, 0 ≤ t ≤ T .
(a) Find the change of measure dQ/dP = Λ so that St is a martingale under
Q. You should show that EΛ = 1. [10 marks]
(b) Give the formula for the price at time t < T of the contract that pays
Y = eST at time T . [5 marks]
(c) Give the self-financing replicating portfolio for the contract in (b). [5
marks]
3. Let Mt = B
2
t −Bt − t, 0 ≤ t ≤ T .
(a) Show that Mt is a martingale. [5 marks]
(b) State what is meant by the Predictable Representation Property of Brow-
nian motion process Bt. Specify this result for the martingale Mt in this
question. [5 marks]
(c) Find a random time change τt so that the process Vt =M(τt) is a Brownian
motion. Justify this by using the theorem of Levy on characterisation of
Brownian motion. [5 marks]
(d) Write Vt in terms of Brownian motion Bt. Show that the process Xt =
V ([M,M ]t) is in fact Mt. [5 marks]
34. Solve the following partial differential equations for the function f(x, t) by
using the probability method. Specify the diffusion process and give its gener-
ator. You may exchange expectation and integration without justification, and
use that the fourth moment ofN(µ, σ2) distribution is given by µ4+6µ2σ2+3σ4.
(a) t∂
2f
∂x2
(x, t) + ∂f
∂t
(x, t) = 0, for 0 ≤ t ≤ T , and f(x, T ) = x4. [10 marks].
(b) t∂
2f
∂x2
(x, t) + ∂f
∂t
(x, t) = −x4, for 0 ≤ t ≤ T , and f(x, T ) = 0. [10 marks].
5. Explain briefly (in one or two sentences and referring to mathematical theo-
rems if possible) the following concepts.
(a) A market model and a claim or option. [5 marks]
(b) Self-financing portfolios and their use in pricing theory. [5 marks]
(c) Arbitrage, and how to check if it exists. [5 marks]
(d) Equivalent martingale measure and its role in pricing options. [5 marks]
End of examination questions
End of examination paper