4404
Simulation IEOR Spring 2008
Midterm Exam
Lecturer: Jose Blanchet
Maximum 70 points
Note that you could get up to 95 points if you solve QUESTION II and the extra-credit
problem. The score DOES NOT accumulate in future exams / assignments but can help
dene boundary grades (e.g. B+ to A- and so forth).
Provide as clear and concise answers as you can for the questions below. Clarity will be
evaluated as part of the grade. The exam is designed so that it should not be computational
intensive. However, if you feel that you are running out of time, a good strategy is to write
step-by-step, in a clear and concise way, how would you answer the question if you had more
time. If your procedure addresses the main issues behind the computational analysis you
might receive partial credit.
1. (20 points) Provide a procedure that can be used to generate a random variable T
with hazard rate function (t) = t + 2t3 . (Be careful with multiple roots if you are using
inversion).
2. Pick one of the following two questions (QUESTION I or QUESTION II) and provide
answers to all the parts in your selected question. You must indicate in your solution book
which question are you addressing as ONLY ONE will be graded. If you fail to CLEARLY
specify which question are you answering the grader will selected one arbitrarily.
QUESTION I (25 points)
Suppose you wish to estimate = Ef (X ) given a function f ( ) for a given random variable
X (in order to avoid technical di culties we assume that f ( ) is bounded). For instance,
X could be the price of a stock at a given time and f ( ) could be the payment prole of a
contract on the underlying stock.
Suppose you have an algorithm that allows you to generate iid copies of X (say, X1 ; X2 ; ...)
and consider the estimator
n
1X
f (Xj ) :
n=
n j =1
Now assume that there exists a function g ( ) (which we may assume bounded) for which you
can compute = Eg (X ) in closed form and consider the family of estimators (depending
1
on a parameter ) dened via
X
1X
en ( ) =
f (Xj ) +
(g (Xj )
n j =1
n j =1
n
n
):
a) (5 points) Is en ( ) an unbiased estimator for ? (i.e. is it the case the E en ( ) = ?).
Are the en ( ) consistent as n % 1? (i.e. is it true that en ( ) ! as n % 1?). Justify
s
your answers.
b) (10 points) Suppose that is xed. Explain how would you compute an approximate
95% condence interval based on en ( ). Provide an explicit expression for the variance of
your variance expression in terms of the variances of f (Xj ), g (Xj ) and their covariance.
c) (10 points) Find the value of (in terms of V ar (f (X1 )) ; V ar (g (X1 )) and cov (f (X1 ) ; g (X1 )))
that minimizes the width of your 95% condence interval.
QUESTION II (30 points)
Assume that (Xn : n
that
0) is a mean zero Gaussian AR(1) process. In other words, assume
Xn+1 = Xn + "n+1 ;
where the "n are iid standard Gaussian random variables, j j < 1 and
s
2
2 (0; 1)
a) points) (5 Find an expression for the transition density (p (x; y ) : x; y 2 R) of the process
(Xn : n 0). In other words, what is
p (x; y ) =
d
P ( X1
dy
y j X0 = x) .
(Note that you don even have to evaluate the previous derivative, just compute what is the
t
density of X1 given X0 = x.)
b) (10 points) Find a constant
such that
2 (0; 1) and a probability density function (v (y ) : y 2 R)
p(x; y )
v (y )
(1)
for all values of x 2 [ 1; 1].
Hint: Note that (a + b)2
2(a2 + b2 ):
c) (5 points) Assume that (1) holds and show that there exists a probability transition
density q (x; y ) such that
p(x; y ) =
v (y ) + (1
)q (x; y ) x 2 [ 1; 1]
:
p (x; y )
x 2 [ 1; 1]
=
2
(2)
d) (10 points) Use equation (2) to explain how to represent the process (Xn : n 0) as a
regenerative process. In order to explain your answer, you can assume that you can simulate
variates with density v ( ) and transitions according to q (x; ) for any x 2 [ 1; 1] and indicate
clearly how to identify the regenerative cycles in your simulation.
3. (25 points) Let (S (t) : t 0) be the price process of an asset (suppose that you can
simulate the vector (S (t1 ), S (t2 ), ..., S (tm )) for any t1 < t2 < ::: < tm ). Let
X = max((S (t1 )
and
Y = max
S (t2 )
m
X
:::
S (tj )=m
j =1
S (tm ))1=m
1; 0)
!
1; 0 :
Provide a step-by-step procedure that can be used to compute an approximate 95% condence interval for the correlation between X and Y based on n iid replications of (S (t1 ); :::S (tm )).
You don have to carry-over all the computations, but you need to clearly explain all the
t
elements in your procedure.
Hint: Note that the correlation between X and Y can be written as a function of the form
g (E (XY ) ; EX 2 ; EY 2 ; EX; EY ).
Note: There is really nothing special about the specic form of X and Y here, I just wanted
you to have a specic example to have in mind that is popular in nance.
4. (Extra-credit: 20 points) Martin has been taking French lessons for some time. Lalli,
who is Martin mother, wishes to test Martin and see how much French does he really
s
know. She bought a huge dictionary with about 3,000 pages and she is thinking to select
some denitions according to the uniform distribution among all the denitions given in the
dictionary, so that she can test what the proportion of denitions that Martin knows in
s
French. Obviously, the procedure has to be easily implemented in few minutes with a pocket
calculator that can generate uniform random variables one at a time.
Design a procedure that allows Lalli achieve this task (i.e. sampling concepts from the
dictionary in such a way that all concepts are equally likely to be sampled. Note that
an obvious, but wrong, procedure would be to sample a page uniformly at random and
then, within that page, a word uniformly at random, but this clearly generates bias towards
concepts that require lengthy explanations, why?).
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