11 Option. Payoffs and Option Strategies. Answers to Questions and Problems

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1 11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of \$80 and a cost of \$5. Graph the profits and losses at expiration for various stock prices. 73

2 74 CHAPTER 11 OPTION PAYOFFS AND OPTION STRATEGIES 2. Consider a put option with an exercise price of \$80 and a cost of \$4. Graph the profits and losses at expiration for various stock prices..

5 ANSWERS TO QUESTIONS AND PROBLEMS 77 The worst outcomes occur when the stock price is very low or very high. First, the strangle loses \$1 for each dollar the stock price falls below \$58. With a zero stock price, the strangle loses \$58. If the stock price is too high, the strangle also loses money. Because the stock could theoretically go to infinity, the potential loss on the strangle is unbounded. For stock prices of \$58 or \$132, the strangle gives exactly a zero profit. 8. Assume that you buy a call with an exercise price of \$100 and a cost of \$9. At the same time, you sell a call with an exercise price of \$110 and a cost of \$5. The two calls have the same underlying stock and the same expiration. What is this position called? Graph the profits and losses at expiration from this position. At what stock price or prices will the position show a zero profit? What is the worst loss that the position can incur? For what range of stock prices does this worst outcome occur? What is the best outcome and for what range of stock prices does it occur? This position is a bull spread with calls, because it is designed to profit if the stock price rises. The entire position has a zero profit if the stock price is \$104. At this point, the call with the \$100 exercise price can be exercised for a \$4 exercise profit. This \$4 exercise value exactly offsets the price of the spread. The worst loss occurs when the stock price is \$100 or below, because the option with the \$100 exercise price cannot be exercised, and the entire position is worthless. This gives a \$4 loss. The best outcome occurs for any stock price of \$110 or above and the total profit is \$6. 9. Consider three call options with the same underlying stock and the same expiration. Assume that you take a long position in a call with an exercise price of \$40 and a long position in a call with an exercise price of \$30. At the same time, you sell two calls with an exercise price of \$35. What position have you created? Graph

6 78 CHAPTER 11 OPTION PAYOFFS AND OPTION STRATEGIES the value of this position at expiration. What is the value of this position at expiration if the stock price is \$90? What is the position s value for a stock price of \$15? What is the lowest value the position can have at expiration? For what range of stock prices does this worst value occur? This is a long position in a butterfly spread. If the stock price is \$90, the value of the spread is zero. For a \$15 stock price, the spread is worth zero. The entire spread can be worth zero at expiration. This zero value occurs for any stock price of \$30 or below and \$40 or above.

7 ANSWERS TO QUESTIONS AND PROBLEMS Assume that you buy a portfolio of stocks with a portfolio price of \$100. A put option on this portfolio has a striking price of \$95 and costs \$3. Graph the combined portfolio of the stock plus a long position in the put. What is the worst outcome that can occur at expiration? For what range of portfolio prices will this worst outcome occur? What is this position called? The worst result is a portfolio value of \$95. The purchase of the put for \$3 gives a loss of \$8. This worst outcome occurs for a terminal stock portfolio value of \$95 or less. This combined position is an insured portfolio. The position insures against any terminal portfolio value less than \$95 or any loss greater than \$8.

8 80 CHAPTER 11 OPTION PAYOFFS AND OPTION STRATEGIES 11. Consider a stock that sells for \$95. A call on this stock has an exercise price of \$95 and costs \$5. A put on this stock also has an exercise price of \$95 and costs \$4. The call and the put have the same expiration. Graph the profit and losses at expiration from holding the long call and short put. How do these profits and losses compare with the value of the stock at expiration? If the stock price is \$80 at expiration, what is the portfolio of options worth? If the stock price is \$105, what is the portfolio of options worth? Explain why the stock and option portfolio differ as they do. No matter what stock price results, the option portfolio will have \$1 less profit than the stock itself. For example, the option portfolio costs \$1, but both options are worthless at a stock price of \$95. Therefore, at a stock price of \$95, the stock has a zero profit, and the option portfolio has a \$1 loss. Further, the option portfolio will be worth exactly \$95 less than the stock at every price. With a stock price of \$80, the call is worthless and the put will be exercised against the option holder for an exercise loss of \$15. Therefore, the option portfolio is worth \$15 for an \$80 stock price. If the stock trades for \$105, the option portfolio will be worth \$ Assume a stock trades for \$120. A call on this stock has a striking price of \$120 and costs \$11. A put also has a striking price of \$120 and costs \$8. A risk-free bond promises to pay \$120 at the expiration of the options in one year. What should the price of this bond be? Explain. A portfolio consisting of one long call, one short put, and a riskless investment equal to the common exercise price of the two options gives exactly the same payoffs as a share of the underlying stock on the common expiration date. This put-call parity relationship requires that this portfolio of long call, short put, plus riskless investment should have the same price as the stock. With our data, the riskless bond must therefore cost \$120 \$11 \$8 \$117. The riskless interest rate must be 2.53 percent.

17 ANSWERS TO QUESTIONS AND PROBLEMS 89 price. The investor would choose to hold a call that expired in one month (the same time until the short position must be covered). For example, assume the investor held a \$100 call that expired in one month. If the stock price at the end of the month was \$105, then the investor would exercise the call and purchase the stock at \$100 to cover the short position. The cost of the protection would be the price of the call option. The other strategy would combine a short put position with the short stock position. In this case, the investor would use the premium income received from writing the put to cover any losses associated with stock price increases. As happens with a covered call position, this premium income would provide limited protection against adverse stock price movements, and would limit potential gains associated with stock price decreases. For example, assume the investor wrote a \$100 put that expired in one month. If the stock price at the end of the month was \$95, then the owner of the put would exercise the put against your client, forcing your client to purchase the stock at \$100. This loss of \$5 on the option contract will offset the \$5 gain in the value of the short stock position. At prices above \$100, the put will expire worthless and the premium income will offset some of the losses in the short stock position.

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