The current price of a stock is $16. In 6 months, the price will be either $20 or $12. The annual risk-free rate is 3%. Find the price of a call option on the stock that has a strike price of $14 and that expires in 6 months. (Hint: Use daily compounding.) Assume a 365-day year. Do not round intermediate calculations. Round your answer to the nearest cent.

Correct Answers: 3.13

Solution

The stock's range of payoffs in six months is $20 - $12 = $8. At expiration, the option will be worth $20 - $14 = $6 if the stock price is $20, and zero if the stock price is $12. The range of payoffs for the stock option is $6 - 0 = $6.

Equalize the range to find the number of shares of stock: Option range/Stock range = $6/$8 = 0.75.

With 0.75 shares, the stock's payoff will be either 0.75($20) = $15.00 or 0.75($12) = $9.00. The portfolio's payoff will be $15.00 - $6.00 = $9.00, or $9.00 - 0 = $9.00.

The present value of $9.00 at the daily compounded risk-free rate is:

PV = $9.00/(1 + (0.03/365))^365/2 = $8.87.

The option price is the current value of the stock in the portfolio minus the PV of the payoff:

V = 0.75($16) - $8.87 = $3.13.