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COMP0048 Financial Engineering
Alternative Assessment
2021
Throughout this examination Wt or W is a standard Brownian motion:
Exam Instructions:
_x000f_
This assessment consists of four questions. You are required to answer
all questions.

The questions are not equally weighted. The total marks are 210.

Complete mathematical steps must be presented to qualify for full
credit.
Examiners attach great importance to legibility, accuracy and clar_xfffe_ity of solutions.

All questions should be neatly/clearly hand written and converted
to pdf as a single document. Any other format will not be accepted.
COMP0048 PLEASE TURN OVER
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1. This question is on di§erential equations arising in nance.
a. [25 Marks] Consider the following IVP consisting of a Kolmogorov
equation for u (x; t); and initial condition
@u
@t =
@
2u
@x2
; u (x; 0) = _x001a_ 0
x
x < 0 x > 0
(1.1)
If N ( ) represents the Normal Cumulative Distribution Function,
Solve (1:1) to show the solution is
u (x; t) = xN p
x
2t

+
r
t

e
_xfffe_
x
2=4t
:
b. [35 Marks] Solve the ordinary di§erential equation
S du
dS +
1
2

2S
2
d
2u
dS2
= _xfffe_ 1;
for the function u (S) with boundary conditions
u (S0) = 0;
u (S1) = 0:

and are constants.
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2. This question is on stochastic calculus.
a. [10 Marks] Consider the stochastic di§erential equation
d (log yt) = (
log yt) dt + dWt
:
The parameters
; ; are constant. Use Itù to obtain dyt
yt
:
b. Which of the following processes are Martingales
(i) [10 Marks] Yt = t
2Wt 2
Z
0
t
sWsds:
(ii) [10 Marks] Gt = e

t sin Wt for
2 R with
2 (0; 1).
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3. This question is on solutions of the Black-Scholes equation
The Black Scholes formula for the value of a put option P (S; t) on a
non-dividend paying stock, is
P (S; t) = SN( d1) + Ee r(T t)N( d2)
From this expression, nd the Black Scholes value of the put option in
the following limits:
(a) [12 Marks] (time tends to expiry) t ! T, > 0;
(b) [18 Marks] (volatility tends to zero) ! 0, t < T; (c) [6 Marks] (volatility tends to in nity) ! 1, t < T; (d) [30 Marks] T ! 1, > 0 and nite. Hint: there are three
cases to consider.
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4. This question is on a two-factor option pricing model
a. [20 Marks] Consider two nancial assets S1 and S2 which evolve
according to the di§usion processes
dS1 = 1S1dt + 1S1dW1
dS2 = 2S2dt + 2S2dW2

1
; 2
; 1; 2 are constants. The Brownian motions W1 and W2
satisfy E [dW1dW2] = 0: By constructing a portfolio of the form
_x0005_ = V 1S1 2S2;
obtain a partial di§erential equation for the fair price of an option
V (S1; S2; t).
b. [34 Marks] An Exchange Option gives the holder the right to
exchange one asset for another, in some ratio. The payo§ for this
contract at expiry is max (q1S1 q2S2; 0); where q1 and q2 are
constants.
By assuming that the solution has the form
V (S1; S2; t) = q1S2U ( ; t)
with = S1=S2; derive a partial di§erential equation for U ( ; t)
and state the nal condition on U ( ; t):
COMP0048 END OF PAPER
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