MATH 294: MATHEMATICS OF FINANCE (WINTER 2018)

HOMEWORK.

Assignment 1, due Tuesday, January 23, 2018, in class.
1. Using a list of option prices taken from the Yahoo Finance website after the close of the stock market on Friday, January 19, 2018, perform a similar calculation to that done in the Example in the text, with Wal-Mart (symbol: WMT) in place of Cisco. For this, use the first WMT call option expiring during the month of February and for which the strike price is strictly greater than the current stock price. 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 by 30% or down by 20% since purchase of the option. Please record the data you used and include all relevant information used. Including a print-out of the option price table that you use to answer this question is requested.
For those who do not have the book yet, here is the text of the Example.
Example. On January 4, 2000, a European call option on Cisco (symbol: CSCO) stock has a price of \$33. The option expires in January, and the strike price is \$70. The price of Cisco stock on January 4 is \$102. If one bought such an option on 100 shares of Cisco, the option would cost \$3,300, and on January 21, 2000 (third Friday of January), one would have the right to buy 100 shares of Cisco at a price of \$70 per share. Suppose for simplicity that \$1 on January 4 is worth \$1 on January 21, 2000.
Scenario 1: Suppose the price of Cisco stock on January 21 is \$120 per share. This current price of the stock is called the spot price of the stock. The holder of the option will exercise it and make a net profit per share of \$120 -\$70 -\$33 (spot price of stock on January 21 - price under exercise of option - option price) and hence a net profit of \$1,700. This is a 1700/33 % = 51.5% profit on the \$3,300 initial investment. On the other hand, if the \$3,300 had been directly invested in stock, the investor could have bought 32 whole shares of stock and the profit would have been \$18 times 32 = \$576 on an investment of \$ 102 times 32 = \$3,264, which is a 57600/3264% = 17.6% profit.
Scenario 2: Suppose the price of Cisco stock on January 21 is \$67 per share. The holder of the option will not exercise it and takes a loss of \$33 per share (the cost of the option per share) and hence a net loss of \$3,300. This is a 100% loss on the \$3,300 initial investment. On the other hand, if the \$3,300 had been invested directly in stock, the loss would have been \$35 times 32 =\$1,120 or a 34.3% loss on an investment of \$3,264 in stock.

2. Suppose that a stock is currently selling for \$50 per share. A forward contract is to be written committing the holder of the long position in the contract to buy 100 shares of stock 3 months from now for \$51 per share. Suppose that a bank is charging interest on short term loans at the rate of 4% per annum (continuously compounded) on a 3-month loan. Describe a strategy for trading in any or all of the forward contract, the stock and a short term loan which creates 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. An 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. Find a formula for the payoff for a strangle where the call has a strike price of K1 and the put has a strike price of K2 and K2 < K1. 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.)

Assignment 2, due Tuesday, February 6, 2018, in class.
From Chapter 2, Exercises 2 (part (d) is optional), 3 (part (d) is optional), 5 (note that you can use the result from Exercise 4 if you wish - there is a typo in the book at the end of Exercise 5, it should read "Exercise 4" rather than "Exercise 3").

One more problem (not optional): Consider the CRR model described in Exercise 2 of Chapter 2 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).

Optional problem: Exercise 4 in the book.

Assignment 3, due Tuesday, February 20, 2018, in class.
Exercises 6, 7 (except (d) parts) from Chapter 2, Exercise 2 from Chapter 3, and the exercise in this pdf file, click here to access it.

Assignment 4, due Thursday, March 1, 2018, in class.
Exercises 3, 4 in Chapter 3.