GS/MATH 6910.03
Stochastic Calculus in Finance

Friday, December 3, 1999

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.

Question 1

Let tex2html_wrap_inline38 be a Brownian motion, and let tex2html_wrap_inline40 be its filtration. Which of the following random variables are measurable with respect to the tex2html_wrap_inline42-field tex2html_wrap_inline44 ?

Question 2

Suppose that tex2html_wrap_inline58 is adapted to a filtration tex2html_wrap_inline40, and that for s<t, tex2html_wrap_inline64 is independent of tex2html_wrap_inline66 and has mean zero and variance t-s. Show that tex2html_wrap_inline58 and tex2html_wrap_inline72 are martingales.

[Note: Since you aren't given that tex2html_wrap_inline58 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 tex2html_wrap_inline76.]

Question 3

Let tex2html_wrap_inline38 be a Brownian motion, and consider the equation
Starting with tex2html_wrap_inline82, compute tex2html_wrap_inline84 for n=1,2,3,4,5.

[Note: Finding tex2html_wrap_inline88 is easy. For the rest, try tex2html_wrap_inline90, tex2html_wrap_inline92, tex2html_wrap_inline94, tex2html_wrap_inline96. Use Itô's lemma to solve for the appropriate constants. You should notice some overlap with problem 3.6 from Assignment 2.]

Question 4

A straddle with expiration date T is a contingent claim based on an underlying stock. The value of the straddle at expiration is tex2html_wrap_inline100, where tex2html_wrap_inline102 is the stock price at expiration, and k and a are parameters. Assume a Geometric Brownian Motion model for the stock price, with tex2html_wrap_inline108 the true drift, tex2html_wrap_inline42 the volatility, r the risk-free rate of return, and tex2html_wrap_inline114 the initial stock price. Write tex2html_wrap_inline38 for the driving Brownian motion under the true model P, and tex2html_wrap_inline120 for the corresponding Brownian motion under the risk-neutral model tex2html_wrap_inline122. Let v denote the initial price of the straddle.

Question 5

Consider the SDE
where tex2html_wrap_inline138 and tex2html_wrap_inline140 are constants. Take tex2html_wrap_inline142 and verify that tex2html_wrap_inline144 solves this SDE, where
tex2html_wrap_inline38 is a Brownian motion, and tex2html_wrap_inline150.

[Remark 1: Recall that tex2html_wrap_inline152 if tex2html_wrap_inline154, and tex2html_wrap_inline156 otherwise. So tex2html_wrap_inline158 and tex2html_wrap_inline160. You may use those formulae if at some point you find you need to show that tex2html_wrap_inline162.]

[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, tex2html_wrap_inline164 is also a Brownian motion (it is a continuous martingale, with tex2html_wrap_inline166). So the SDE for tex2html_wrap_inline168 is really the Cox-Ingersoll-Ross model for interest rates. It turns out that its solutions tex2html_wrap_inline168 are always tex2html_wrap_inline172 (which fixes one important defect of certain other interest rate models). One purpose of the above is to demonstrate this in the case tex2html_wrap_inline142. In addition, it turns out that for tex2html_wrap_inline176, the solutions tex2html_wrap_inline168 can actually reach the value 0. The above explicit formula can be used to show this too, as follows, again in the case tex2html_wrap_inline142. It is easy to show that tex2html_wrap_inline184 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 tex2html_wrap_inline188, in which case also tex2html_wrap_inline190.]