Question

You are attempting to value a call option with an exercise price of $145 and one year to expiration. The underlying stock pays no dividends, its current price is $145, and you believe it has a 50% chance of increasing to $155 and a 50% chance of decreasing to $135. The risk-free rate of interest is 5%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model. Answer is not $4.76

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to 155and a 50% chance of decreasing to 135. The two possible stock prices are: S+ = 155 and S- = 135. Therefore, since the exercise price is 145, the corresponding two possible call values are: Cu = 10 and Cd = 0

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (10 - 0)/(155 - 135) = 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) = 145 -2C0 and the certain end-of-year value is 135

Step 4: Calculate the present value of 135 with a one-year interest rate of 5%: 135/(1+0.05)^1 = 128.57

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

145 - 2C0 = 128.57

Step 6: Solve for the value of the call: C0 = 8.21