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
