# YORK UNIVERSITY

### GS/MATH 6910.03 Stochastic Calculus in Finance FINAL EXAMINATION

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 be a Brownian motion, and let be its filtration. Which of the following random variables are measurable with respect to the -field ?
• (a)
• (b)
• (c)
• (d)
• (e)
• (f)

### Question 2

Suppose that 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 .]

### Question 3

Let be a Brownian motion, and consider the equation

Starting with , 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.]

### 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 , 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.
• (a) Express v as a discounted risk-neutral expectation of some explicit function of and the model parameters.
• (b) Using a Girsanov transformation, express v as a discounted expectation of some explicit function of and the model parameters, this time under the true probability P.

### Question 5

Consider the SDE

where 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 .]