UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE COMP0048
ASSESSMENT COM P0048A 7PA
PATTERN
MODULE NAME Financial Engineering
LEVEL: Postgraduate
DATE 24 April 2019
TIME 10:00
TIME ALLOWED 2 hrs 30 mins
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
2018-19
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
Hall Instructions
Standard Calculators
Non-Standard
Calculators
TURN OVER
Financial Engineering, COMP0048 (A7P)
Main Summer Examination Period, 2018-19
Suitable for cohorts: 2018/19
There are FIVE questions. Marks will be awarded for FOUR questions. If more than FOUR are
attempted, then only marks for the best FOUR will count.
Each question is worth 25 marks. The marks avalilable for each part of each question are given
in square brackets.
Standard calculators are permitted.
Throughout this examination Wt or W (t) is a standard Brownian motion.
1. a. [5 Marks] The moment generating function (MGF) of Y, denoted
COMP0048
My ( 0) is given by the expectation
provided the expectation exists. Explain how this definition can
be adapted to calculate moments of different orders.
b. [10 Marks] If Y is a normally distributed random variable, Y rv
N (µ, 0"2 ), then show that the MGF My (0) is
You are required to show all your working when evaluat­
ing the integral.
c. [10 Marks] Consider the stochastic differential equation
for the diffusion Xt. The parameters a, b are constant. Using Ito's
lemma and suitable integration over [O, T], show that
XT = Xo exp { (a - ½b 2) T + b¢v'T}
where¢ rv N (0, 1).
1 TURNOVER
2. a. [10 Marks]
COMP0048
1. Explain what happens to the number of shares and the share
price when a n-m stock split is announced; you may assume
n > m. Give four reasons why a company may issue a stock­
split?
11. Discuss dividends in the context of shares. Your discussion
should include all key dates during the cycle of a dividend
payment.
b. [10 Marks]
i. Explain what happens when you short sell some shares.
11. What is the limiting behaviour of
( r )nt
lim 1 + -
n----t(X) n
Full working must be shown.
c. [5 Marks] In the FX markets, define the terms base currency and
counter currency. What is the meaning of the following quoted
currency pair?
EUR/USD = 1.25492
If at a later time this pair appears on another exchange as
USD /EUR = 0.85543
what is the affect on EUR?
2 CONTINUED
3. a. [10 Marks] Assume that an asset price S evolves according to
the SDE
COMP0048
dS
S = (µ - D) dt + O"dWt,
where µ and O" are constants. In particular S pays out a continu­
ous dividend stream equal to DSdt during the infinitesimal time
interval dt, where D the dividend yield is constant.
Now suppose a European style derivative security is written on
this asset with the properties that at expiry the holder receives
the asset and prior to expiry the derivative pays a continuous cash
flow C (S, t) dt during each time interval of length dt.
Show that the option price satisfies the following partial differen­
tial equation
av 1 2 2 a2v av
8t + 2(]" S as2 + (r - D) Sas - rV = -C (S, t). (3.
1)
b. [15 Marks] In (3.1) suppose that the cash flow C (S, t) has the
form C (S, t) = f (t) S. By writing V = ¢ (t) S and assuming a
final condition at time T given by
V(S,T) = S,
show that the delta of the derivative security is
� (S, t) = e-D(T-t) + 1
T
e-D(r-t) f ( T) dT .
3 TURNOVER
4. a. Consider the pricing equation for the value of a derivative security
COMP0048
V (S, t),
where S 2: 0 is the spot price of the underlying equity, 0 < t::; T
is the time, r 2: 0 the constant rate of interest, and O' is the
constant volatility of S.
The variables ( t, S) can be written as
t = m6t O ::; m ::; M; S = n6S O ::; n ::; N,
where (5t, 5S) are fixed step sizes in turn. V (S, t) is written dis­
cretely as Vnm. An Explicit Finite Difference Method is to be
developed to solve ( 4.1) using a backward marching scheme.
i. [6 Marks] Derive a difference equation in the form
where an, bn, cn should be defined; you may use the following
as a starting point,
v:m _ v:m-1 av v:m v:m n n n+I - n-1
' 8S "" 25S ;
Vn""'- 1 - 2vnm + Vn+I
5S2
11. [5 Marks] Obtain expressions for the final and boundary con­
ditions in finite difference form for a European Call Option.
b. [8 Marks] A European call option is to be priced. Outline a
Monte Carlo method to do this.
c. [6 Marks] Give four advantages of the Monte Carlo method for
pricing derivatives
4 CONTINUED
5. a. [8 Marks] Suppose the spot interest rate r, which is a function
COMP0048
of time t, satisfies the stochastic differential equation
dr = u (r, t) dt + w (r, t) dWt .
The Bond Pricing Equation for a security V = V (r, t; T) is
av 1 2 a2v av
8t
+
2w ar2 + (u - Aw) ar - r
V = 0. (5.1)
By considering an unhedged bond and the risk free return, explain
how and why A arises.
You are not required to derive (5.1)
b. [17 Marks] The extended Hull and White model has risk-adjusted
drift and diffusion, in turn as
u -Aw = rJ ( t) --yr, w (r, t) = c,
where rJ (t) is an arbitrary function of time and 'Y and care con­
stants. Deduce that the value of a zero coupon bond with the
usual redemption value Z (r, T ; T) = 1, is given by
where
and
A (t; T)
Z (r, t; T) = exp (A (t; T) - rB (t; T)),
B (t; T) = _!_ (1 - e--y(T-t))
'Y
-1
T
TJ(T)B (T;T)dT + 2�2 ((T-t) + �e--y(T-t) - 2\e-
2
-y(T-t) -2�).
5 END OF PAPER

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