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Answer :
a. The range of S is 50 (130 - 80) and the range of P is 30 (110 - 80). The hedge ratio of the put is -0.5.
b. The nonrandom payoff to the portfolio is 3S_T - 5P, and the present value of the portfolio depends on the specific values of S_T and P.
c. The value of the put can be solved using the put option pricing formula with the given data.
d. The value of the call option can be calculated using the put-call parity theorem and comparing it to the value of the put option.
a. To show the range of S and P across the two states:
- Range of S: S can take on values of 130 and 80, so the range is 50 (130 - 80).
- Range of P: We can use the put-call parity theorem to determine the range of P. The put-call parity is given by C - P = S - X/(1 + r), where C is the value of the call option. Rearranging the equation, we have P = C - S + X/(1 + r). Substituting the values, we get:
P = C - 130 + 110/1.10 = C - 130 + 100 = C - 30
So the range of P is 30 (C - 30).
The hedge ratio of the put can be calculated by taking the difference in option values and dividing it by the difference in stock prices. Using the given values:
Hedge ratio = (P(130) - P(80)) / (S(130) - S(80))
= (P(130) - P(80)) / (50)
= (P(130) - P(80)) / 50
b. The nonrandom payoff to the portfolio of three shares of stock and five puts can be calculated as follows:
- Stock payoff: 3 * (130 - 100) = 90
- Put payoff: 5 * max(110 - 130, 0) = 0
- Total payoff: 90 + 0 = 90
To find the present value of the portfolio, we discount the payoff by the risk-free rate. Since the risk-free rate is not provided in the question, we cannot calculate the present value.
c. To solve for the value of the put, we need to calculate the option value at each state and take the expected value, weighted by the probabilities of the states.
- Put value at S = 130: P(130) = max(X - S, 0) / (1 + r) = max(110 - 130, 0) / 1.10 = 0
- Put value at S = 80: P(80) = max(X - S, 0) / (1 + r) = max(110 - 80, 0) / 1.10 = 27.27
Expected value of the put: P = (P(130) * P(S = 130)) + (P(80) * P(S = 80))
= (0 * P(S = 130)) + (27.27 * P(S = 80))
d. The value of a call option with an exercise price of 110 can be calculated similarly to the put option. However, since the question does not provide the probabilities of the two states or the risk-free rate, we cannot determine the exact value of the call option.
To verify the put-call parity theorem, we compare the put value (from part c) with the call value. The put-call parity states that C - P = S - X/(1 + r). Since we cannot calculate the call value, we cannot directly verify the put-call parity in this case.
Learn more about probabilities here:
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