Question

Consider a two-state call option valuation problem given the current stock price $240 and the two possibilities for the change in the price are $270 and $170. Also, the strike price is $250 and the risk-free rate is 10%. What is the hedge ratio of the call? b) Calculate the value of a 1-year call option using discrete compounding.

Answer 1

a

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to 270and a 50% chance of decreasing to 170. The two possible stock prices are: S+ = 270 and S- = 170. Therefore, since the exercise price is 250, the corresponding two possible call values are: Cu = 20 and Cd = 0

Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (20 - 0)/(270 - 170) = 0.2

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 - 5C0) = 240 -5C0 and the certain end-of-year value is 170

Step 4: Calculate the present value of 170 with a one-year interest rate of 10%: 170/(1+0.1)^1 = 154.55

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

240 - 5C0 = 154.55

Step 6: Solve for the value of the call: C0 = 17.09