Question

Use the binomial options pricing model to find the price of a call option with a strike price of $40 and one year to expiration. The current stock price is $40 and has equal probabilities for a price of $70 or $30 at expiration in one year. The one year continuous risk-free interest rate is 6%.

A.) What is the hedge ratio for the call option?

B.) What is the price of the call option?

Answer 1

A

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to 70and a 50% chance of decreasing to 30. The two possible stock prices are: S+ = 70 and S- = 30. Therefore, since the exercise price is 40, the corresponding two possible call values are: Cu = 30 and Cd = 0

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (30 - 0)/(70 - 30) = 0.75

B

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 - 1.33333333333333C0) = 40 -1.33333333333333C0 and the certain end-of-year value is 30

Step 4: Calculate the present value of 30 with a one-year interest rate of 6%: 30/(1+0.06)^1 = 28.3

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

40 - 1.33333333333333C0 = 28.3

Step 6: Solve for the value of the call: C0 = 8.77