MATH 194: INTRODUCTION TO THE MATHEMATICS OF FINANCE (WINTER 2012)
Homework is due in the TA's homework box by Friday at 8pm. The homework box is in the basement of AP&M. To access the homework solutions, you will need the login and password given for the summary notes. Be sure to explain your answers, do not just write down an answer or formula.
HOMEWORK 1: Due Friday, January 9, 2009.
1. Suppose that stock in the Widget Company is currently selling for $30 per share. A (long position in a) forward contract is available to buy 100 shares of the stock 3 months from now for $30.10 per share. Suppose that a bank is offering a 3-month CD with an interest rate of 3% per annum (continuously compounded). Describe a strategy for creating an arbitrage profit and establish the amount of the profit.
2. Using a list of option prices (e.g., from the Yahoo finance website), perform a similar calculation to that done in the Example in the web notes, with Verizon (symbol: VZ) in place of Cisco. For this, use the third Verizon call option expiring in this month (January 2012) for which an ``ask" price is listed. You should assume that a European call option for 100 shares of stock is purchased and that at expiration there are two possible scenarios for the stock price: it has gone up or down by 10% since purchase of the option. Make sure to indicate all of the information on which your calculation is based: the date and time on which you looked up prices, the price for the stock itself (use the last closing price if the market is not open when you look up prices), the strike price for the option, the ask price for which you can buy the option, etc.
3. A put option on a stock has a strike price of $100 and is currently selling for a price of $3 with 14 days remaining until expiration. What is the payoff of the option at expiration? Draw a graph of this payoff as a function of the stock price at expiration. If a buyer purchases the put option today and holds it until expiration, what must the stock price be at expiration in order for the buyer to make a net profit?

For solutions to homework 1, click here.

HOMEWORK 2, due Friday, January 16, 2009.
1. 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 and 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 S(T) of the underlying asset. (Make sure to label your axes on the graph.)

2. Suppose that initially (at time zero) you take a long position in a forward contract for 10 shares of a stock where the forward price is $30 per share. At the same time, you sell 10 derivatives of a certain type for $25 each. At the terminal time T, each of these derivatives gives the buyer of the derivative one share of the stock if the stock price per share S(T) at time T is greater than $40, or it pays the buyer of the derivative $20 if S(T) is at or below $40.
(a) Find a formula and draw a graph for the payoff at time T of the derivative as a function of the final stock price S(T).
What is your total profit/loss if
(b) the stock price is $50 at time T,
(c) the stock price is $35 at time T?

3. A company and a bank enter into a 5 year interest rate swap agreement with a notional amount of $200 million. Interest rate payments are made annually at the end of years 1 through 5. The company agrees to make variable interest rate payments each year at the rate of the 12-month LIBOR + 0.2% where the LIBOR rate is recorded at the beginning of the year for which the interest is to be paid. For instance, the variable interest rate applied at the end of the first year on the $200 million will be the 12-month LIBOR rate at the beginning of that year plus 0.2%. The bank agrees to make fixed rate payments each year at a flat rate of 5% per year. Suppose that the 12-month LIBOR rates at the beginning of years 1, 2, 3, 4, 5 are 3.703%, 7.24%, 5.196%, 5.954%, 5.774%, respectively (these are actual numbers for a period in the 1990s). Determine the amounts of the variable interest payments owed by the company at the end of years one through five. What is the net profit or loss over the 5 years for the company?

4. Consider a three period (T=3) binomial model with initial stock price equal to 400, d=0.5, u=3, r=0.2, 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)).
Note: You don't actually need r for this problem, but a future problem will be based on this situation where r will be needed.

For solutions to homework 2, click here.

HOMEWORK 3, due Friday, January 23, 2009.
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. Exercise 1 of Chapter 2 from the web notes (page 28).

For solutions to homework 3, click here.

HOMEWORK 4: Due Friday, January 30, 2009.
1. Exercise 2 (except part (d)) from Chapter 2 of web notes.
2. Exercise 3 (except part (d)) from Chapter 2 of web notes and where for part (c) you should suppose that the European contingent claim is initially priced $1 above the arbitrage free price.
3. Consider the binomial model described in Exercise 4 of Homework 2.
(a) Find the arbitrage free initial price for the European contingent claim described in part (e) of that problem. That European contingent claim is an example of a look back option. (Make sure to recall (that is write down) the necessary information from your solution to Exercise 4 on Homework 2. In particular, make sure to draw the binary tree with stock values and European contingent claim payoffs shown on it.)
(b) Find (alpha(3), beta(3)), the allocations to stock and bond that need to be made over the time period (2,3] as part of a hedging strategy for the European contingent claim. Note that this is a pair of random variables and so you will need to describe the values the pair can take for all possible outcomes.
(c) Use your answer to (b) to explain why you cannot use a recombining binary tree in finding the hedging strategy for this "lookback" option.
4. Use an Excel spreadsheet to find the arbitrage free initial price for a European put option based on a CRR (binomial) model with T=5, S(0)=$100, u=2, d=0.6, r=0.05 when the strike price is K = $110.
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.

For solutions to homework 4, click here.
For excel spreadsheet to go with solutions to homework 4, click here.

HOMEWORK 5: Due Friday, February 6, 2009.
Homework 5 is in the pdf file which can be found by clicking here.
For solutions to homework 5, click here.

MIDTERM and Solutions.

HOMEWORK 6, Due Monday, February 23, 2009, 6pm. The TA will have an extra office hour on Monday, February 23, 2-4 p.m.
It is recommended that you start the following problems early. The homework assignment is in the pdf file obtained by clicking here.
Note for problem 3 that in Excel you can find values of the cumulative normal distribution function Phi.
For problem 4 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).
For solutions to homework 6, click here.
For the two excel spreadsheets that go with the homework solutions, click here and here.

HOMEWORK 7, Due Friday, February 27, 2009.
1. Consider a three-period binomial model with S(0) = 600, u=2, d=1/4, r=0.2, p=1/5.
(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) Let tau be the first time t that S(t) is greater than 650, or, if there is no such time, let tau equal 3. Write out what tau is as a function on Omega. Prove that tau is a stopping time.
(d) Let eta be the last time t that S(t) is greater than 550. Write out what eta is as a function on Omega. Is eta a stopping time? You must give an argument (i.e., proof) to support your answer.
2. Exercises 6(a), 6(b) from Chapter 2 of the webnotes.
3. Write a paragraph summarizing the contributions that Black, Scholes and Merton made to options pricing. In doing this, try to distinguish the roles of Black and Scholes versus Merton. There are several articles on the class webpage that may help you (under the heading Wikepedia and other links). Make sure to cite any sources that you use.
For solutions to homework 7, click here.

HOMEWORK 8, Due Friday, March 6, 2009.
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. Write a one page summary of what you learned from reading the Blinder article on the class website and from the visit by Dr. Constantin Megiris to the class on Friday, February 27.
For solutions to homework 8, click here. For the spreadsheet that goes with this homework, click here.

HOMEWORK 9, Due Friday, March 13, 2009.
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, but don't hand it in.
For solutions to homework 9, click here.