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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 S9.1. Figure S9.1 Profit from long position in Problem 9.9 Problem 9.10 Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option. Ignoring the time value of money, the seller of the option will make a profit if the stock price at maturity is greater than $52.00. This is because the cost to the seller of the option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the seller of the option makes a profit even though the option is exercised. The profit from the short position is as shown in Figure S9.2.Figure S9.2 Profit from short position in Problem 9.10 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 S9.4 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 40TS− and the call option provides no payoff. The options cost $7 and so the total profit is 33TS−. 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 45TS − and the put option provides no payoff. Taking into account the $7 cost of the options, the total profit is 52TS −. 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 S9.4 Profit from trading strategy in Problem 9.12 Problem 10.10 What is a lower bound for the price of a two-month European put option on a non-dividend-paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per annum? The lower bound is 0 05 2 1265 58 6 46e$−. ×/−=. Problem 10.11 A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur? The present value of the strike price is 0 12 4 1260 57 65e$−. ×/= .. The present value of the dividend is 0 12 1 120 80 0 79e−. ×/.=.. Because 5 64 57 65 0 79< − . −. the condition in equation (10.8) is violated. An arbitrageur should buy the option and short the stock. This generates 64 5 59$−=. The arbitrageur invests $0.79 of this at 12% for one month to pay the dividend of $0.80 in one month. The remaining $58.21 is invested for four months at 12%. Regardless of what happens a profit will materialize. If the stock price declines below $60 in four months, the arbitrageur loses the $5 spent on the option but gains on the short position. The arbitrageur shorts when the stock price is $64, has to pay dividends with a present value of $0.79, and closes out the short position when the stock price is $60 or less. Because $57.65 is the present value of $60, the short position generates at least 64 57 65 0 79 5 56$− . −. = . in present value terms. The present value of the arbitrageur’s gain is therefore at least 5 56 5 00 0 56$. −. = ..If the stock price is above $60 at the expiration of the option, the option is exercised. The arbitrageur buys the stock for $60 in four months and closes out the short position. The present value of the $60 paid for the stock is $57.65 and as before the dividend has a present value of $0.79. The gain from the short position and the exercise of the option is therefore exactly equal to 64 57 65 0 79 5 56$− . −. = .. The arbitrageur’s gain in present value terms is exactly equal to5 56 5 00 0 56$. −. = .. Problem 10.12 A one-month European put option on a non-dividend-paying stock is currently selling for 2 50$ .. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum. What opportunities are there for an arbitrageur? In this case the present value of the strike price is 0 06 1 1250 49 75e−. ×/= .. Because 2 5 49 75 47 00.< . − . the condition in equation (10.5) is violated. An arbitrageur should borrow $49.50 at 6% for one month, buy the stock, and buy the put option. This generates a profit in all circumstances. If the stock price is above $50 in one month, the option expires worthless, but the stock can be sold for at least $50. A sum of $50 received in one month has a present value of $49.75 today. The strategy therefore generates profit with a present value of at least $0.25. If the stock price is below $50 in one month the put option is exercised and the stock owned is sold for exactly $50 (or $49.75 in present value terms). The trading strategy therefore generates a profit of exactly $0.25 in present value terms. Problem 11.10 Suppose that put options on a stock …

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