Question

We will derive a two-state call option value in this problem. Data: Se 290; X 180 300; 1 r= 1.1. The two possibilities for sT are 330 and a. The range of S is 150 while that of Cis 30 across the two states. What is the hedge ratio of the call? (Round your answer to 2 decimal places.) Hedge ratio b. Calculate the value of a call option on the stock with an exercise price of 300. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.) (Do not round intermediate calculations. Round your answer to 2 decimal places.) Call value

Answer 1

a). Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $330 and a 50% chance of decreasing to $180.

The two possible stock prices are:

S+ = $330 and S– = $180. Therefore, since the exercise price is $300, the corresponding two possible call values are:

Cu= $30 and Cd= $0.

Calculate the hedge ratio:

(Cu– Cd)/(uS0– dS0) = (30 – 0)/(330 – 180) = 30/150 = 1/5, or 0.20

b). Form a riskless portfolio made up of one share of stock and five written calls. The cost of the riskless portfolio is:

(S0– 5C0) = 300 – 5C0

and the certain end-of-year value is $180.

Calculate the present value of $180 with a one-year interest rate of 10%:

$180/1.10 = $163.64

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

$300 – 5C0 = $163.64

5C0 = $300 - $163.64

C0 = $136.36 / 5 = $27.27