Question

You are attempting to value a call option with an exercise price of $90 and one year to expiration. The underlying stock pays no dividends, its current price is $90, and you believe it has a 50% chance of increasing to $125 and a 50% chance of decreasing to $55. The risk-free rate of interest is 7%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model.

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to 125and a 50% chance of decreasing to 55. The two possible stock prices are: S+ = 125 and S- = 55. Therefore, since the exercise price is 90, the corresponding two possible call values are: Cu = 35 and Cd = 0

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (35 - 0)/(125 - 55) = 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) = 90 -2C0 and the certain end-of-year value is 55

Step 4: Calculate the present value of 55 with a one-year interest rate of 7%: 55/(1+0.07)^1 = 51.4

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

90 - 2C0 = 51.4

Step 6: Solve for the value of the call: C0 = 19.3