MATH 194: INTRODUCTION TO THE MATHEMATICS OF FINANCE (WINTER 2007)
Homework is due at the beginning of section.
HOMEWORK 1: Due Wednesday, January 17, 2007.
1. Exercise 1 from Chapter 1 of the web notes.
2. Suppose that a stock is currently selling for $20 per share. A (long position in a) forward contract is available to buy 100 shares of stock 3 months from now for $20.20 per share. Suppose that a bank is offering interest at the rate of 5% per annum (continuously compounded) on a 3-month deposit. Describe a strategy for creating an arbitrage profit and establish the amount of the profit.
3. A combination option called a strangle is obtained by taking a long position in a (European) call and a (European) put option with the same expiration date but differing strike prices, all based on the same underlying asset. A strangle is similar to a straddle in that the investor who buys the strangle is betting that there will be a large movement in the price of the underlying, but is uncertain whether it will involve an increase or a decrease in the price. Typically the price has to move further for the investor to make a profit from a strangle, but the downside risk is typically less than with a straddle. Find a formula for the payoff for a strangle where the put has a strike price of K1 and the call has a strike price of K2 and K1 < K2. Draw a graph of this payoff as a function of the final price of the underlying asset. (Make sure to label your axes on the graph.)

HOMEWORK 2: Due Wednesday, January 24, 2007.
1. Consider the multiperiod binomial model with 0< d< u. Show that this primary market model is not viable (that is, there is an arbitrage opportunity) if either (a) 1+r is less than or equal to d, or (b) 1+r is greater than or equal to u.
Hint: in each case, describe a self-financing trading strategy that is an arbitrage opportunity and prove that it is indeed an arbitrage opportunity.
2. Consider a three period (T=3) binomial model with initial stock price equal to 100, d=0.5, u=3, r=0.1, p=1/3.
(a) Draw the (non-recombining) binary tree illustrating the possible paths followed by the stock price process.
(b) By labelling the tree, describe the sample space Omega of all possible outcomes (paths for the stock price process).
(c) Indicate the probability associated with each individual member of Omega.
(d) Describe all events (collections of outcomes) that can be distinguished knowing just the value of the stock price at time zero and time one (you should make sure to include the empty set and the whole of Omega in this description, but what other events are included?).
(e) Indicate in a separate color on your binary tree the final values (one for each path) of a European contingent claim whose value at T=3 is X= max(S(0), S(1), S(2), S(3)).
3. Exercise 1 of Chapter 2 from the web notes (page 28).

HOMEWORK 3: Due Wednesday, January 31, 2007.
1. Exercise 2 (except part (d)) from Chapter 2 of web notes.
2. Exercise 3 (except part (d)) from Chapter 2 of web notes.
3. Use an Excel spreadsheet to find the arbitrage free initial price for a European put option based on a CRR model with T=5, S(0)=$100, u=2, d=0.8, r=0.1 when the strike price is K = $120.
Please be aware that the Excel spreadsheet uses a recombining binary tree and so when using the formula for the initial arbitrage free price, in computing probabilities associated with a final value of the put, you need to compute the probability associated with each path leading to that final value. There may be more than one such path for a given final value in the recombining tree.

HOMEWORK 4: Due Wednesday, February 7, 007.
1. For this problem click here.
2. Consider the CRR model described in Exercise 2 of Chapter 2 of the web notes as the model for a stock and a bond. A forward contract is to be offered under which the holder of a long position in the forward contract will buy 100 shares of stock at time T=2 for a fixed price F. (Here F is the total amount to be paid for the 100 shares). Remember that no money changes hands at time zero when a forward contract is written. What value should F take in order that there is no arbitrage opportunity for the investor who holds a long or a short position in the contract? Explain your reasoning fully (in particular, identify an associated European contingent claim and derive the value of F using arbitrage pricing for the contingent claim).

MIDTERM: Wednesday, February 14, 2007
The midterm will cover the material treated in the lectures, notes, reading and related homework up through February 7, 2007. In particular, this includes the in-class lectures, the printed notes available on the webpage, and homework assignments 1-4. Students should bring a blue book, calculator, pens and pencils, and their I.D. The midterm will be held at the usual class meeting time of 4pm on Wednesday, February 14 in WLH 2005 (note that this is NOT the regular class room). Please make sure to sit with at least one seat between you and the next person in the exam room.

HOMEWORK 5, Due Wednesday February 21, 2007.
There will be no homework assignment due on February 14, 2007 due to the midterm that day. It is recommended that you start the following problems as soon as the material is covered in class. In particular, attempting parts (a) and (b) of problem 1 may provide some additional practice for the midterm.
1 and 2. For these problems, click here.
3. Exercise 6, parts (a) and (b) from Chapter 2 of the web notes.

HOMEWORK 6, Due Wednesday, February 28, 2007.
1. Exercise 6(c) from Chapter 2 of the web notes.
2. Use an Excel spreadsheet (following the "Computing American option trees" handout) to find the initial arbitrage free price for an American put based on a binomial model with T = 5, S(0) = $125, u = 1.3, d = 0.96, r = 0.05, where the strike price is K = $120. (Note that your tree will be recombining.)
3. For this exercise, click here.
Correction to original problem 3: please use data from January 18, 2007 through February 22, 2007. If you have used data from January 16, 2007 through February 22, 2007, that is ok.
Hints for problem 3: Note that T is the number of time periods, which is one less than the number of data points that you have (which will be for times t=0,1, ...., T). With pricing of the American put, you may find that the agreement with the real price is not high (percentage wise). Note however that the real price of the put is small in actual terms. Why do you think there might be a difference - note that we have made certain assumptions in our model, including no transaction costs? If you cannot find the closing prices for options on Friday, February 23, you can use $2.50 for the price of the call on February 23 and $0.05 for the price of the put.

HOMEWORK 7, Due Wednesday, March 7, 2007.
For this homework, click here. Note that in Excel you can find values of the cumulative normal distribution function Phi.

HOMEWORK 8, Due Wednesday, March 14, 2007. Extension: students can hand in the homework by 6pm on Thursday, March 15, 2007. Students should take the homework to the TA's office by that time.
Exercises 3 and 4 at the end of Chapter 3 of the web notes.
You may also try Exercise 2 in Chapter 3 to practise for the final. If you hand in Exercise 2, it will count for extra credit. However, you do not need to hand it in.

For solutions to Homework 7 click here and for Homework 8, click here. You will need the password for this class in order to access these solutions. (They are also available from the web notes page.)