Problem 9.1. An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor's profit with the stock price at the maturity of the option. The investor makes a profit if the price of the stock on the expiration date is less than $37. In these circumstances the gain from exercising the option is greater than $3. The option will be exercised if the stock price is less than $40 at the maturity of the option. The variation of the investor's profit with the stock price in Figure S9.1. Figure S9.1 Investor's profit in Problem 9.1 Problem 9.2. An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor's profit with the stock price at the maturity of the option. The investor makes a profit if the price of the stock is below $54 on the expiration date. If the stock price is below $50, the option will not be exercised, and the investor makes a profit of $4. If the stock price is between $50 and $54, the option is exercised and the investor makes a profit between $0 and $4. The variation of the investor's profit with the stock price is as shown in Figure S9.2. Figure S9.2 Investor's profit in Problem 9.2 Problem 9.9. Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option. Ignoring the time value of money, the holder of the option will make a profit if the stock price at maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the option is exercised, but the holder of the option takes a loss overall. The profit from a long position is as shown in Figure S9.3. Figure S9.3 Profit from long position in Problem 9.9 Problem 9.12. A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader's profit with the asset price. Figure S9.6 shows the variation of the trader's position with the asset price. We can divide the alternative asset prices into three ranges: a) When the asset price less than $40, the put option provides a payoff of and the call option provides no payoff. The options cost $7 and so the total profit is . b) When the asset price is between $40 and $45, neither option provides a payoff. There is a net loss of $7. c) When the asset price greater than $45, the call option provides a payoff of and the put option provides no payoff. Taking into account the $7 cost of the options, the total profit is . The trader makes a profit (ignoring the time value of money) if the stock price is less than $33 or greater than $52. This type of trading strategy is known as a strangle and is discussed in Chapter 11. Figure S9.6 Profit from trading strategy in Problem 9.12 Problem 9.13. Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date. The holder of an American option has all the same rights as the holder of a European option and more. It must therefore be worth at least as much. If it were not, an arbitrageur could short the European option and take a long position in the American option. Problem 9.14. Explain why an American option is always worth at least as much as its intrinsic value. The holder of an American option has the right to exercise it immediately. The American option must therefore be worth at least as much as its intrinsic value. If it were not an arbitrageur could lock in a sure profit by buying the option and exercising it immediately. Problem 9.17. Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in four months. Explain how the terms of the option contract change when there is a) A 10% stock dividend b) A 10% cash dividend c) A 4-for-1 stock split a) The option contract becomes one to buy shares with an exercise price . b) There is no effect. The terms of an options contract are not normally adjusted for cash dividends. c) The option contract becomes one to buy shares with an exercise price of . Problem 9.23. The price of a stock is $40. The price of a one-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options. Draw a diagram illustrating how the investor's profit or loss varies with the stock price over the next year. How does your answer change if the investor buys 100 shares, shorts 200 call options, and buys 200 put options? Figure S9.7 shows the way in which the investor's profit varies with the stock price in the first case. For stock prices less than $30 there is a loss of $1,200. As the stock price increases from $30 to $50 the profit increases from –$1,200 to $800. Above $50 the profit is $800. Students may express surprise that a call which is $10 out of the money is less expensive than a put which is $10 out of the money. This could be because of dividends or the crashophobia phenomenon discussed in Chapter 19. Figure S9.8 shows the way in which the profit varies with stock price in the second case. In this case the profit pattern has a zigzag shape. The problem illustrates how many different patterns can be obtained by including calls, puts, and the underlying asset in a portfolio. Figure S9.7 Profit in first case considered Problem 9.23 Figure S9.8 Profit for the second case considered Problem 9.23