Question

You are attempting to value a call option with an exercise price of $108 and one year to expiration. The underlying stock pays no dividends, its current price is $108, and you believe it has a 50% chance of increasing to $130 and a 50% chance of decreasing to $86. The risk-free rate of interest is 10%. Calculate the call option’s value using the two-state stock price model. (Do not round intermediate calculations and round your final answer to 2 decimal places.)

Call options' value:

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to 130and a 50% chance of decreasing to 86. The two possible stock prices are: S+ = 130 and S- = 86. Therefore, since the exercise price is 108, the corresponding two possible call values are: Cu = 22 and Cd = 0

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (22 - 0)/(130 - 86) = 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) = 108 -2C0 and the certain end-of-year value is 86

Step 4: Calculate the present value of 86 with a one-year interest rate of 10%: 86/(1+0.1)^1 = 78.18

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

108 - 2C0 = 78.18

Step 6: Solve for the value of the call: C0 = 14.91