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 $145 and a 50% chance of decreasing to $71. The risk-free rate of interest is 9%. Calculate the call option's value using the two-state stock price model.

Answer 1

I have answered the question below

Please up vote for the same and thanks!!!

Do reach out in the comments for any queries

Answer:

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $145 and a 50% chance of decreasing to $71. The two possible stock prices are: S+ = $145 and S- = $71. Therefore, since the exercise price is $108, the corresponding two possible call values are: Cu = $37 and Cd = $0.

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (37 - 0)/(145 - 71) = 0.50

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 $71

Step 4: Calculate the present value of $71 with a one-year interest rate of 9%: $71/1.09 = $65.1376

Step 5: Set the value of the hedged position equal to the present value of the certain payoff:
$108 - 2C0 = $65.1376

Step 6: Solve for the value of the call: C0 = $21.4312