Question

You are attempting to value a call option with an exercise price of $65 and one year to expiration. The underlying stock pays no dividends, its current price is $65, and you believe it has a 50% chance of increasing to $90 and a 50% chance of decreasing to $40. The risk-free rate of interest is 8%. Based upon your assumptions, calculate your estimate of the call option's value using the two-state stock price model. (Do not round intermediate calculations. Round your answer to 2 decimal places.)  

Value of the call            $    

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $90 and a 50% chance of decreasing to $40.

The two possible stock prices are:

S+ = $90 and S– = $40. Therefore, since the exercise price is $65, the corresponding two possible call values are:

Cu= $90 - $65 = $25 and Cd= $40 - $65 = $0.

Step 2: Calculate the hedge ratio:

(Cu– Cd)/(uS0– dS0) = (25 – 0)/(90 – 40) = 25/50 = 0.5

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

(S0 – 2C0) = 65 – 2C0

and the certain end-of-year value is $40.

Step 4: Calculate the present value of $40 with a one-year interest rate of 8%:

$40/1.08 = $37.04

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

$65 – 2C0= $37.04

2C0 = $65 - $37.04

C0 = $27.96 / 2 = $13.98