The exam is to be handed in to me, or slipped under my office door (N625 Ross), before noon on Monday December 6. You are to work on the problems individually, without discussing them with any other person. You may freely consult your class notes, or the textbook. If you have a question about the interpretation of any of the problems, you may send me e-mail (to saltnexus.yorku.ca). I will be checking e-mail periodically over the weekend. With the exception of the first problem, you must justify your reasoning. Do all five questions.
be a Brownian motion, and let 
be its filtration.
Which of the following random variables are measurable with respect to the
-field 
?
is adapted to a filtration , and that for
s<t, 
is independent of 
and has mean zero and variance t-s.
Show that and 
are martingales.
[Note: Since you aren't given that is continuous, you
should not try to use Itô's lemma or quadratic variation here. Instead you
should work directly with the properties of conditional expectations, being sure
to tell me which such properties you use. To get started, observe that
.]
be a Brownian motion, and consider the equation
,
compute 
for n=1,2,3,4,5.
[Note: Finding 
is easy. For the rest, try 
,
, 
, 
. Use
Itô's lemma to solve for the appropriate constants. You should notice some
overlap with problem 3.6 from Assignment 2.]
, where 
is the stock price at expiration, and k and a are
parameters.
Assume a Geometric Brownian Motion model for the stock price,
with 
the true drift,
the volatility,
r the risk-free rate of return, and
the initial stock price. Write for the driving Brownian motion under
the true model P, and 
for the corresponding Brownian motion under
the risk-neutral model 
. Let v denote the initial price of the
straddle.
and the model parameters.
and the model parameters,
this time under the true probability P.
and 
are constants. Take 
and verify that
solves this SDE, where
is a Brownian motion, and 
.
[Remark 1: Recall that 
if 
, and 
otherwise. So 
and 
. You may use those formulae if
at some point you find you need to show that
.]
[Remark 2: There is a point to this problem, which I'm now going to
explain, though I don't want you to work through any of this additional
material for the exam. First of all, 
is also a Brownian motion (it
is a continuous martingale, with
). So the SDE for 
is really the Cox-Ingersoll-Ross
model for interest rates. It turns out that its solutions are always 
(which fixes one important defect of certain other interest rate models). One
purpose of the above is to demonstrate this in the case . In
addition, it turns out that for
, the solutions can actually reach the value 0. The
above explicit formula can be used to show this too, as follows, again in the
case .
It is easy to show that
has a normal distribution, so can take positive and negative values. In
fact, it can be positive at some times, and negative at others. From this it
follows that there will be times
t at which 
, in which case also 
.]