Question

We will derive a two-state call option value in this problem. Data: Se = $160, X= $170, 1+/=110. The two possibilities for Sy are $190 and $110. The portfolio consists of 1 share of stock and 4 calls short.

Required:
a. The range of Sis $80 while that of C is $20 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 $170. (Do not use continuous compounding to calculate the present value of Xin 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. The hedge ratio of the call = (Change in the option price) / (Change in the stock price) = (C2 - C1) / (S2 - S1) = ($20 - $-20) / ($190 - $110) = 0.8.

b. Using the hedge ratio computed in part (a), the value of a call option on the stock with an exercise price of $170 can be calculated in both states. In the high state, the stock price is $190 and the call option would be exercised for a payoff of $20. In the low state, the stock price is $110 and the call option would not be exercised, giving a payoff of $0. Then, the expected call payoff is:

(0.5 × $20) + (0.5 × $0) = $10

The present value of the expected payoff using the effective interest rate is:

$10 / (1 + 1.1) = $4.54

Therefore, the value of the call option is $4.54.