Problem 21-32 You would like to be holding a protective put position on the stock of XYZ Co. to lock in a guaranteed minimum value of $100 at year- end. XYZ currently sells for $100. Over the next year the stock price will increase by 13% or decrease by 13%. The T-bill rate is 4%. Unfortunately, no put options are traded on XYZ Co. a. Suppose the desired put option were traded. How much would it cost to purchase? (Do not round intermediate calculations and round your final answer to 2 decimal places.) Cost to purchase b. What would have been the cost of the protective put portfolio? (Do not round intermediate calculations and round your final answer to 2 decimal places.) Cost of the protective put portfolio c. What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective put with X = 100? Show that the payoff to this portfolio and the cost of establishing the portfolio match those of the desired protective put. (Do not round intermediate calculations and round your final answer to 2 decimal places.) S = 87 S= 113 Portfolio Buy 0.5 shares Invest in T-bills Total

(a) We know, the hedge ratio can be calculated by the following formula :-

Hedge Ratio :- H = (P_u - P_d) / (uS₀ - dS₀)

where, P_u = put option's value when the stock price goes up / increases

P_d = put option's value when the stock price goes down / decreases

uS₀ & dS₀  are the stock prices in two states

Here, X = $100;

If the price of stock rises by 13% in one year, the stock price would be $100 * (1+0.13) = $113. Since the strike price of the put option is $100, it will be $0, if the stock price rises by 13%.

uS₀ = $100 *(1 + 0.13) = $113 ; P_u = 0

If the stock price falls by 13% in one year, the stock price would be  $100 * (1-0.13) = $87. Since the strike price of the put option is $100 and the price of the stock would be $87, the put option will have an intrinsic value of $13.

dS₀ = $100 *(1 - 0.13) = $87 ; P_d = $13

Therefore, Hedge ratio :- H = (P_u - P_d) / (uS₀ - dS₀)

Or, H = (0 - 13) / (113 - 87)

= -13 / 26

= -1/2

Thus, Hedge ratio :- H = - 0.5

Hence, a portfolio comprised of 1 share & 2 puts provides a guaranteed payoff of $113, with present value :-

$113 / (1 + 0.04)¹ = $113 / 1.04 = $108.6538

Thus, S + 2P = $108.6538

Or, $100 + 2P = $108.6538

Or, 2P = $108.6538 - $100

Or, 2P = $8.6538

Or, P = $4.3269 or, P = $4.33 (rounded off)

Therefore, the cost to purchase a desired put option = $4.33

(b) The cost of a protective put portfolio with $100 guaranteed payoff :-

Cost of a protective put portfolio = $100 + $4.33 = $104.33

(c) As per the above given condition we have to achieve a goal where the exposure to the stock would be same as to the hypothetical protective put portfolio. Since, the put's hedge ratio is - 0.5, the portfolio consists of (1 - 0.5) = 0.5 shares of stock $50 (as, 1 share = $100) and the reamaning funds ($104.33 - $50) = $54.33 invested in T-bills earning 4% interest.

This payoff is identical to that of the protective put portfolio. Hence, it can be concluded that the combined portfolio strategy of stock & T-bills replicates both the cost & payoff of the protective put portfolio.