The price of Chive Corp. stock will be either $86 or $119 at the end of the year. Call options are available with one year to expiration. T-bills currently yield 5 percent. a. Suppose the current price of the company's stock is $97. What is the value of the call option if the exercise price is $85 per share? (Do not round intermediate calculations and round your answer to 2 decimal places, e. g., 32.16.) b. Suppose the exercise price is $115 and the current price of the company's stock is $97. What is the value of the call option now? (Do not round intermediate calculations and round your answer to 2 decimal places, e. g., 32.16.)

Answer and Explanation:

a). Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $119 and a 50% chance of decreasing to $86.

The two possible stock prices are:

S + = $119 and S - = $86. Therefore, since the exercise price is $85, the corresponding two possible call values are:

Cu = $34 and Cd = $1.

Step 2: Calculate the hedge ratio:

(Cu - Cd) / (uS0 - dS0) = (34 - 1) / (119 - 86) = 33/33 = 1

Step 3: Form a riskless portfolio made up of one share of stock and one written calls. The cost of the riskless portfolio is:

(S0 - C0) = 97 - C0

and the certain end-of-year value is $86.

Step 4: Calculate the present value of $86 with a one-year interest rate of 5%:

$86/1.05 = $81.90

Step 5: Set the value of the hedged position equal to the present value of the certain payoff:

$97 - C0 = $81.90

C0 = $97 - $81.90 = $15.10

b). Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $119 and a 50% chance of decreasing to $86.

The two possible stock prices are:

S + = $119 and S - = $86. Therefore, since the exercise price is $115, the corresponding two possible call values are:

Cu = $4 and Cd = $0.

Step 2: Calculate the hedge ratio:

(Cu - Cd) / (uS0 - dS0) = (4 - 0) / (119 - 86) = 4/33

Step 3: Form a riskless portfolio made up of four shares of stock and thirty three written calls. The cost of the riskless portfolio is:

(4S0 - 33C0) = 4 (97) - 33C0 = 388 - 33C0

and the certain end-of-year value is $86.

Step 4: Calculate the present value of $86 with a one-year interest rate of 5%:

$86/1.05 = $81.90

Step 5: Set the value of the hedged position equal to the present value of the certain payoff:

$388 - 33C0 = $81.90

33C0 = $388 - $81.90

C0 = $306.10 / 33 = $9.28