UNIVERSITY OF DUBLIN
BU7527
TRINITY COLLEGE
FACULTY
OF
S CIENCE
SCHOOL OF MATHEMATICS
M.Sc. (Finance) Degree Examination
M ATHEMATICS
Wednesday 14th December, 2011
Michaelmas Term, 2011
OF
C ONTINGENT C LAIMS
GMB
Dr. Michael Peardon
9:30–11:30am
Prof. Angela J. Black
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 equal marks
You may not start this exam, until you are instructed to do so by the invigilator.
Page 2 of 4
1.
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(a) A game is played by rolling a fair dice and scoring the number shown. Any time
the dice lands on six, it is rolled again and the next number added to the score.
This is repeated until the dice does not land on six. Find the expected value of
the score of the game.
(b) Each quarter, a forecaster predicts whether a company will either return a profit,
return a loss or go bankrupt. The forecaster correctly predicts quarters with a
loss with 60% probability, profitable quarters with 75% probability and bankrupcies with 20% probability. In any given quarter, there is a 50% probability a
company is in profit, a 45% probability it makes a loss and so a 5% probability
it goes bankrupt. The forecaster predicts the company will make a profit next
quarter. What is the probability it will in fact go bankrupt?
(c) The losses of mobile phones by a staff-member of a small company occur obey
Poisson statistics, and the rate of loss is one phone per month on average.
i. Describe what this means for the dependence on k of pk the probability one
phone will be lost next month given k were lost this month.
ii. Find the probability more than three phones will be lost next month.
iii. In three months, five phones were lost. Find the probability exactly four
were lost in a single one of the months.
c UNIVERSITY OF DUBLIN 2011
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Page 3 of 4
2.
(a) A random number Z ∈ [−1, 1] has probability density function given by
3
fZ (z) = (z + 1)2
8
Find its expected value and variance.
(b) Two random numbers, A and B are drawn from the uniform distribution in the
range [0, 1]. The numbers are redrawn until A + B < 1. Find the probability
density function for A alone.
(c) Two random numbers X > 0 and Y > 0 are drawn from the joint probability
density
fXY (x, y) = ye−y(x+1)
i. Find fY , the density function for Y alone
ii. Find P (Y > 1)
3.
(a) Briefly describe how a Monte Carlo estimator of an integral can be constructed,
paying attention to what can be said about the statistical properties and uncertainty of the estimator.
(b) A Monte Carlo estimate of
Z
1
I=
xe0.1x dx
−1
is to be computed. Five random numbers {U1 , U2 , . . . U5 } are drawn from a
uniform-variate random number generator with Ui ∈ [0, 1]. The values of these
random numbers are given in Table 1 (see next page). Use these values to
estimate I , giving an estimate of the 95%-confidence interval for your answer.
You may assume the estimator obeys the central limit theorem.
(c) Using the same ensemble of random numbers, construct an antithetic estimator
for I along with an estimate for the 95%-confidence interval.
c UNIVERSITY OF DUBLIN 2011
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Table 1: Sample random numbers (for use in question 3.)
4.
k
Uk
1
0.612
2
0.919
3
0.077
4
0.789
5
0.987
(a) In a game, a score X is either doubled with probability α or halved with probability 1 − α.
i. For what value of α is X0 , X1 , X2 , . . ., the sequence of values of X as the
game is played a martingale?
ii. The game starts with initial value X0 = 4 and stops at time T when either
XT = 1 or XT = 16. Find
A. E[XT ]
B. P (XT = 1), the probability the game ends with X = 1.
(b)
i. What properties define a Markov process?
A garage repairs a particular fault in a car in two steps; diagnosis and repair.
In one day, there is a probability 1/5 that a car will arrive to be serviced. If
the factory is busy the car is sent away, so the system can be described by
three states {χ0 , χ1 , χ2 }, corresponding to the shop being idle, diagnosing
a fault or repairing a fault respectively. In the diagnosis stage, there is a
probability 3/10 the fault is diagnosed in the day and moved on to the repair
stage. The repair stage completes with probability 1/10 each day.
ii. Write a Markov matrix describing the stochastic transitions the system can
make each day.
iii. Show that the long-time average probabilities for the garage to be in each
of the three states are given by P (χ0 ) = 2/11, P (χ1 ) = 3/11, P (χ2 ) = 6/11
respectively.
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