HOMEWORK.

Assignment 1, due Wednesday, January 16, 2008, in class.
1. Chapter 1, exercise 1.
Using a list of option prices (e.g., from the website for the Chicago Board Options Exchange or Yahoo Finance), perform a similar calculation to that done in the Example in the text, with Intel (symbol: INTC) in place of Cisco. For this, use the first Intel call option expiring in the current month and for which a 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 40% since purchase of the option.
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 $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. 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 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.)

Extra credit project: This is a challenging project. You do not need to do this project. If you do it well, it will count for extra credit.
Consider a finite market model that is viable and complete. Show that there is a minimal superhedging strategy for any American contingent claim in this setting. (Hint: you may need to use the martingale representation property.) Use your analysis to find the initial arbitrage free price for an American contingent claim. Part of your answer will involve showing that this price is indeed arbitrage free.