The current price of a stock is $20. In 1 year, the price will be either $24 or $16. The annual risk-free rate is 4%. Find the price of a call option on the stock that has a strike price of $21 and that expires in 1 year. (Hint: Use daily compounding.) Assume a 365-day year. Do not round intermediate calculations. Round your answer to the nearest cent.

Correct Answer: 1.74

Solution

The stock's range of payoffs in one year is $24 - $16 = $8. At expiration, the option will be worth $24 - $21 = $3 if the stock price is $24, and zero if the stock price is $16. The range of payoffs for the stock option is $3 - 0 = $3.

Equalize the range to find the number of shares of stock: Option range/Stock range = $3/$8 = 0.375.

With 0.375 shares, the stock's payoff will be either $9.00 or $6.00. The portfolio's payoff will be $9.00 - $3.00 = $6.00, or $6.00 - 0 = $6.00.

The present value of $6.00 at the daily compounded risk-free rate is:

PV = $6.00/(1 + (0.04/365))³⁶⁵ = $5.76.

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

V = 0.375($20) - $5.76 = $1.74.