UNIVERSITY OF DUBLIN
XBU75271
TRINITY COLLEGE
FACULTY
OF
A RTS , H UMANITIES
S CIENCES
AND
S OCIAL
SCHOOL OF BUSINESS
M.Sc. (Finance) Degree Examination
Michaelmas Term Examination
M ATHEMATICS
Friday 14th December, 2012
Michaelmas 2012
OF
C ONTINGENT C LAIMS
LUCE LOWER
09:30–11:30
Dr. Michael Peardon
Closed Book Examination
Non-programmable calculators are permitted for this examination – please indicate the
make and model of your calculator on each exam book used.
The examination is two hours in duration
Attempt THREE questions. All questions carry 20 marks. Marks for each part of the
question are given in square brackets
You may not start this exam, until you are instructed to do so by the invigilator.
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XBU75271
1.
(a) A die is rolled and six coins are flipped. What is the probability the number of
coins landing on heads is equal to the number shown on the die?
[6 marks]
(b) Three dice are rolled. Find the probability the first number rolled was a four,
given the total on the three dice was 12.
[8 marks]
(c) A model predicts there is a 75% probability an online trading company will see
a rise in profits when GDP grows. Conversely, there is a 10% probability profits
will rise when GDP falls. In any year, there is a 65% probability GDP will grow.
The company sees a rise in profits in 2013. What is the probability GDP grew
that year?
[6 marks]
2. Two random numbers X and Y have a joint distribution function
fXY (x, y) =



1
3
(x + y)
when 0 ≤ x ≤ 2, 0 ≤ y ≤ 1
0
otherwise
Find
(a) The expected value and variance of X .
[5 marks]
(b) Cov(X, Y ) = E[XY ] − E[X]E[Y ].
[5 marks]
(c) P (X > 1|Y < 21 ).
[5 marks]
(d) P (X > Y ).
[5 marks]
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XBU75271
3.
(a) Explain how the Monte Carlo method can be used to estimate an integral. Describe the circumstances that enable a statistically reliable estimate of the uncertainty in a Monte Carlo estimator to be quoted.
[4 marks]
(b) Using the samples of a uniform variate random number U presented in Table 1,
estimate
2 √
Z
e
I=
x
dx
0
making sure to quote an estimate of the uncertainty in your result.
[8 marks]
(c) Explain what is meant by a control variate. Under what conditions are they
useful.
Using the same dataset (in Table 1), evaluate the Monte Carlo estimator Iˆ for
the same integral but constructed using g(x) = 1 +
√
x as a control variate.
[8 marks]
Table 1. Five samples of uniform-variate random numbers.
k
Uk
1
0.2251
2
0.3859
3
0.8488
4
0.4983
5
0.8073
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XBU75271
4.
(a) Define briefly what is meant by
i. A Markov process
[2 marks]
ii. A Martingale
[2 marks]
(b) Xk is a sequence of random numbers, such that
Xk+1

 2X
k
=
 X /2
k
with probability p
with probability (1 − p)
i. Find the value of p that makes X a martingale.
[4 marks]
ii. The sequence starts with value X1 = 4 and stops at time T when either
XT = 1 or XT = 32. Find P (XT = 1).
[4 marks]
(c) A system has three states and makes transitions according to a Markov process
with Markov matrix

1
2

1 0




M = 0 0 1 


1
0
0
2
Find the long-time average the system will be in each of its three states.
[8 marks]
c UNIVERSITY OF DUBLIN 2012
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