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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 40 option provides no payoff The options cost 7 and so the total profit is 33 and the call TS TS 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 TS 45 and the put option provides no payoff Taking into account the 7 cost of the options the total profit is 52 TS 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 58 e 65 6 46 0 05 2 12 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 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 80 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 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 is therefore at least 5 56 5 00 in present value terms The present value of the arbitrageur s gain The arbitrageur invests 0 79 of this at 12 for one month to The present value of the dividend is Because 0 12 4 12 e 57 65 0 56 0 79 59 0 12 1 12 60 e 5 56 The arbitrageur s gain in present value terms is exactly equal to 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 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 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 …

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