Assignment #5: Options

1. A European put option and a European call option with an exercise price of $45 will expire in two months. They sell for $2.65 and $5.32, respectively. If the stock is currently priced at $47.30, what is the annual continuously compounded rate of interest?

2. Rob wishes to buy a European put option on BioLabs, Inc., a non-dividend–paying common stock, with a strike price of $40 and six months until expiration. BioLabs’ common stock is currently selling for $30 per share, and Rob expects that the stock price will either rise to $60 or fall to $15 in six months. Rob can borrow and lend at the risk-free effective annual rate of 8 percent. (Note: The effective annual rate of 8% is different from the stated annual rate of 8%. The stated annual rate can be divided based on the compounding periods. The effective annual rate cannot be divided.)

a. What should the put option sell for today?

b. If no options currently trade on the stock, is there a way to create a synthetic put option with identical payoffs to the put option just described? If there is, how would you do it?

c. How much does the synthetic put option cost? Is this greater than, less than, or equal to what the actual put option costs? Does this make sense?

3. Maverick Manufacturing, Inc., must purchase gold in three months for use in its operations. Maverick’s management has estimated that if the price of gold were to rise above $875 per ounce, the firm would go bankrupt. The current price of gold is $815 per ounce. The firm’s chief financial officer believes that the price of gold will either rise to $975 per ounce or fall to $740 per ounce over the next three months. Management wishes to eliminate any risk of the firm going bankrupt. Maverick can borrow and lend at the risk-free effective annual rate of 6.50 percent.

a. Should the company buy a call option or a put option on gold? To avoid bankruptcy and to minimize the cost of hedging, what strike price and time to expiration would the company like this option to have?

b. How much should such an option sell for in the open market?

4. A company’s asset is worth 20 now, and will be worth 16 (in the bad state) or 24 (in the good state) with equal probabilities tomorrow. The riskfree interest rate is 10%. The company lives for only one period. It promises to repay 18 to the debtholders next period.

a) What is the value of debt?

b) What is the expected yield of the debt? How does it compare with the riskfree rate?

c) What is the value of equity?

d) Suppose the company receives a government loan guarantee. What is the loan guarantee worth? What is the value of the company after it receives the loan guarantee? Why is the company worth more?

5. Consider an American put option written on a non-dividend paying stock. The strike price is 50. The put will expire in two years. The stock price follows a two-period binomial as in the graph. At each date, the natural probability that the stock price increases is one half. The stock is currently selling at 50. A two-year annual coupon bond with annual coupon rate of 10% is selling at par.

18 48 128 30 80 50

(a) What is the price of the American put option?

(b) What is the price of an American call option written on the same stock with the same strike price and expiration date?

(c) Prove or disprove: the put-call parity holds for American puts and American calls.

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