HOMEWORK 1: Due Friday, January 13, 2012, 8pm.
1. Suppose that stock in an oil 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.
(a) Suppose the stock price in 3 months time is denoted by S(3) (this is a random variable). Write an expression for the payoff to the holder of a long position in the contract after the 3 months have passed? (The payoff is the final value of the contract to its holder.)
Suppose further that a bank is offering a 3-month CD with an interest rate of 3% per annum (continuously compounded).
What is the payoff (total value=principal plus interest) of such a CD at the end of the 3-months, where the principal amount is $3000?
(c) Given the three instruments (stock, forward contract and CD), describe a strategy involving holding long or short positions in these instruments for creating an arbitrage profit and establish the amount of the profit.

2. A put option for a share of a certain technology stock has a strike price of $80 and is currently selling for a price of $5 with 14 days remaining until expiration. Assuming that S(14) is the price of the stock at 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 net profit, you will want to take into account the price paid for the option as well as the payoff).

3. 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 Apple (symbol: AAPL) in place of Cisco. For this, use the third Apple 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, the expiration date, etc.

Solutions to homework 1 (you will need the class notes password to access these).

HOMEWORK 2, due Friday, January 20, 2012, noon. Note the change in time for homework to be due because of the change in class schedule.
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.) Suppose the initial price of the strangle is L. Write a formula for the net profit for the strangle and draw its graph as a function of S(T).

2. A type of derivative sells for $40 at time zero. At the terminal time T, this derivative gives the buyer of the derivative one share of ABC stock if the stock price per share S(T) at time T is greater than $60, or it pays the buyer of the derivative $20 if S(T) is at or below $60.
(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).
(b) Write a formula for the net profit (which is a negative number if there is a net loss) for the buyer of this derivative. (This formula will be a function of S(T)).
Now suppose that you SELL 10 of these derivatives at time zero.
(c) Write a formula for your net profit by time T.
(d) What is your net profit if the stock price is $70 at time T.
(e) What is your net profit if the stock price is $50 at time T?

3. A company and a bank enter into a 5 year interest rate swap agreement with a notional amount of $100 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 $100 million will be the 12-month LIBOR rate at the beginning of that year plus 0.2% (this is the same as twenty basis points; 1 basis point = 1/100 %). The bank agrees to make fixed rate payments at the end of each year at a flat (simple interest) 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. In return they will receive the 5% interest payments from the bank at the end of each year. What is the net profit or loss over the 5 years for the company? (See the webnotes page for an example of an actual swap agreement entered into by a bank and a company in the 1990s.)
Click here for homework 2 solutions.

HOMEWORK 3, due Friday, January 27, noon.
1. 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= min(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.
2. Consider the single period (T=1) 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.
3. Exercise 1(a), (b) of Chapter 2 from the web notes (page 28).
4. Consider the same single period binomial model as in Exercise 1 of Chapter 2. 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 =1 for a fixed price of F per share. Remember that no money changes hands at time zero when a forward contract is written, so the initial price of the contract is always zero. The forward price F provides flexibility in pricing of the contract. 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 free pricing for the contingent claim).

Click here for homework 3 solutions.

HOMEWORK 4, due Friday, February 3, noon.
1. Exercise 1(c), (d) of Chapter 2 from the web notes (page 28). Also, describe a hedging strategy for the put option described in 1(d), Chapter 2 of the webnotes.
2. Exercise 2 (except part (d)) from Chapter 2 of web notes.
3. 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.

Click here for homework 4 solutions. (Corrected).

HOMEWORK 5, due Friday, February 10, noon.
1. Consider the binomial model described in Problem 1 of Homework 3.
(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 Problem 1 on Homework 3. 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.
2. 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.
See the handout on the webnotes page for how to setup an Excel spreadsheet for this. Please be aware that the Excel spreadsheet will use 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.
Problems 3 and 4 for Homework 5 are in the pdf file which can be found by clicking here.

Click here for homework 5 solutions (corrected). An excel spreadsheet to go with Problem 2 is here.

There will be no homework assignment due on Friday, February 17.
Solutions to the midterm are here (you will need the same password as for the homework solutions to access these).

HOMEWORK 6, due Friday, February 24, 2012, noon.
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).

Click here for corrected homework 6 solutions. An excel spreadsheet to go with Problem 4 is here.

HOMEWORK 7, due Friday, March 2, 2012, noon.
1. Consider a three-period binomial model with S(0) = 500, u=2, d=1/5, r=0.2, p=1/6.
(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 600, 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 450. 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), 6(c) from Chapter 2 of the webnotes.
3. 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.)

Click here for homework 7 solutions. An excel spreadsheet to go with Problem 3 is here.

HOMEWORK 8, due Friday, March 9, 2012, noon.
Exercises 3 and 4 from Chapter 3 of the webnotes.
Click here for homework 8 solutions.

HOMEWORK 9, due Friday, March 16, 2012, noon. This is the last homework assignment for the quarter.
Exercise 2 from Chapter 3 of the webnotes.
Click here for homework 9 solutions.