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3. (Portfolio Insurance) Put options are also known as portfolio insurance because their pay-off is inversely related to the stock price. This exercise is to illustrate this practical use of put options. Suppose on January 12, 2022 we have 10,000 shares of the S&P 500 index with a total worth of $47, 250, 000. Although we think the price will move upwards, we want to limit potential losses in April 2022 by buying a number of Put options with April 14, 2022 maturity and strike 4, 670. Price of this option was $141.75. Buying 10,000 Put options thus costs $1,417,500 and can be financed either by selling 300 shares of the index or by borrowing with zero interest rate (to simplify). We consider two portfolios: 1. Original portfolio P1 : 10,000 shares of the S&P 500 index; 2. 'Insured' portfolio P2 : 9,700 shares of the S&P 500 index and 10,000 Put option contracts as defined above; 3. 'Insured' by debt portfolio P3: 10,000 shares of the S&P 500 index, 10,000 Put option contracts and a debt of $1,417,500. a. What is the value of each portfolio on January 12, 2022? b. Let ST be the future random price of the S&P 500 index on April 14, 2022. Express the values of each portfolio on this day in terms of St. c. (Insurance). Show that on April 14, 2022, the value of the insured portfolio P2 will be at least (10,000 300) x 4, 670 - $45, 299,000 in all cases; loss of at most 4.129%. What is the minimum possible value of P3 on that day ? d. (Cost of Insurance). What would be the values of each portfolio, if the index value ST = 4,800? = = 3. (Portfolio Insurance) Put options are also known as portfolio insurance because their pay-off is inversely related to the stock price. This exercise is to illustrate this practical use of put options. Suppose on January 12, 2022 we have 10,000 shares of the S&P 500 index with a total worth of $47, 250, 000. Although we think the price will move upwards, we want to limit potential losses in April 2022 by buying a number of Put options with April 14, 2022 maturity and strike 4, 670. Price of this option was $141.75. Buying 10,000 Put options thus costs $1,417,500 and can be financed either by selling 300 shares of the index or by borrowing with zero interest rate (to simplify). We consider two portfolios: 1. Original portfolio P1 : 10,000 shares of the S&P 500 index; 2. 'Insured' portfolio P2 : 9,700 shares of the S&P 500 index and 10,000 Put option contracts as defined above; 3. 'Insured' by debt portfolio P3: 10,000 shares of the S&P 500 index, 10,000 Put option contracts and a debt of $1,417,500. a. What is the value of each portfolio on January 12, 2022? b. Let ST be the future random price of the S&P 500 index on April 14, 2022. Express the values of each portfolio on this day in terms of St. c. (Insurance). Show that on April 14, 2022, the value of the insured portfolio P2 will be at least (10,000 300) x 4, 670 - $45, 299,000 in all cases; loss of at most 4.129%. What is the minimum possible value of P3 on that day ? d. (Cost of Insurance). What would be the values of each portfolio, if the index value ST = 4,800? = =

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