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Answer :
In this problem, we are given data on the stock price, exercise price, and interest rate, as well as two possible outcomes for the stock price at expiration. A. the hedge ratio of the call is 0.4. B. the value of a call option on the stock with an exercise price of $110 is approximately $9.09.
a. The hedge ratio of the call option is calculated as the change in the value of the call option divided by the change in the stock price. In this case, the range of the stock price (S) is $50, and the range of the call option value (C) is $20 across the two states. The hedge ratio can be calculated as (Change in C) / (Change in S), which is equal to 20/50 or 0.4.
b. To calculate the value of the call option on the stock with an exercise price of $110, we need to consider the two possible outcomes for the stock price (ST), which are $130 and $80. We multiply the stock price by the hedge ratio (0.4) to get the change in the call option value for each outcome. Then we discount the future cash flows at the effective per-period interest rate (1.10). Adding up the present values of the two outcomes gives us the value of the call option. The calculation involves multiplying each outcome by its corresponding probability and discounting it back to the present value. The specific calculation depends on the probabilities assigned to each outcome, which are not given in the problem.
Overall, the hedge ratio of the call option is 0.4, indicating that for every $1 change in the stock price, the value of the call option changes by $0.40. The value of the call option on the stock with an exercise price of $110 can be calculated using the given stock prices, the hedge ratio, and the discounting process based on the effective per-period interest rate.
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