Question

We will derive a two-state put option value in this problem. Data: S₀ =100;X=110;1+r= 1.10. The two possibilities for Sₜ are 130 and 80.

a. Show that the range of S is 50 while that of P is 30 across the two states. What is the hedge ratio of the put?
b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
c. What is the present value of the portfolio?
d. Given that the stock currently is selling at 100 , show that the value of the put must be 10.91

Answer 1

The two-state put option value is derived using given data, with the range of stock price being 50 and put value being 30, a hedge ratio of -0.6, a nonrandom portfolio payoff of 390, and a present value of 354.55. The price of each put is deduced to be 10.91.

Step-by-step explanation:

We will derive a two-state put option value in this problem using the given data: S₀ = 100; X = 110; 1+r = 1.10. The two possibilities for Sₜ are 130 and 80.

a. The range of the stock price (S) is 50, calculated as the difference between the two possible future stock prices (130 - 80). For the put option, the range of P is 30, which represents the difference in put values between the two states. The put value in each state is the maximum of either 0 or the strike price minus the stock price. Hence, when Sₜ is 130, P is 0 (since 110 - 130 is negative, and the option wouldn't be exercised), and when Sₜ is 80, P is 30 (110 - 80). The hedge ratio (also known as the delta) for the put can be calculated as the change in the put value divided by the change in the stock price, which is -30/50 or -0.6.

b. A portfolio consisting of three shares of stock and five puts will have the following payoffs: If Sₜ is 130, the stock value will be 3 * 130 = 390 and put value will be 5 * 0 = 0, so the total payoff is 390. If Sₜ is 80, the stock value will be 3 * 80 = 240 and the put value will be 5 * 30 = 150, so the total payoff is 390. Thus, the portfolio payoff is nonrandom, always equaling 390.

c. The present value of the portfolio is calculated by discounting the future payoff (390) by the interest rate. Therefore, it is 390 / 1.10 = 354.55.

d. Given that the stock price (S₀) is currently 100, we can use a no-arbitrage argument and the put-call parity to show that the value of the put must be 10.91. Starting with the present value of the portfolio (354.55), we subtract the cost of purchasing three shares at the current price (3 * 100 = 300), resulting in 54.55. This is the total cost of purchasing five puts. Thus, each put must cost 54.55 / 5 = 10.91.