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20 12 17 the put with the higher exercise price must

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17. The put with the higher exercise price must cost more. Therefore, the net outlay to establish the portfolio is positive. Position S T < 90 90 S T 95 S T > 95 Write put, X = $90 –(90 – S T ) 0 0 Buy put, X = $95 95 – S T 95 – S T 0 Total 5 95 – S T 0 The payoff and profit diagram is: 0 S T Payoff 5 90 95 Profit Net outlay to establish position 18. Buy the X = 62 put (which should cost more but does not) and write the X = 60 put. Since the options have the same price, your net outlay is zero. Your proceeds at maturity may be positive, but cannot be negative. Position S T < 60 60 S T 62 S T > 62 Buy put, X = $62 62 – S T 62 – S T 0 Write put, X = $60 –(60 – S T ) 0 0 Total 2 62 – S T 0 0 S T 2 60 62 Payoff = Profit (because net investment = 0) 20-13
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19. According to put-call parity (assuming no dividends), the present value of a payment of $85 can be calculated using the options with April maturity and exercise price of $85. PV(X) = S 0 + P – C PV($85) = $83.20 + $2.45 – $0.95 =$84.70 20. The following payoff table shows that the portfolio is riskless with time-T value equal to $10: Position S T 10 S T > 10 Buy stock S T S T Write call, X = $10 0 –(S T – 10) Buy put, X = $10 10 – S T 0 Total 10 10 Therefore, the risk-free rate is: ($10/$9.50) – 1 = 0.0526 = 5.26% 21. From put-call parity: C – P = S 0 – X/(l + r f ) T If the options are at the money, then S 0 = X and: C – P = X – X/(l + r f ) T The right-hand side of the equation is positive, and we conclude that C > P. 22. a., b. Position S T < 100 100 S T 110 S T > 110 Buy put, X = $110 110 – S T 110 – S T 0 Write put, X = $100 –(100 – S T ) 0 0 Total 10 110 – S T 0 The net outlay to establish this position is positive. The put you buy has a higher exercise price than the put you write, and therefore must cost more than the put that you write. Therefore, net profits will be less than the payoff at time T. 20-14
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0 S T 110 100 10 Payoff Profit c. The value of this portfolio generally decreases with the stock price. Therefore, its beta is negative. 23 a. Joe’s strategy Position Cost Payoff S T 400 S T > 400 Stock index 400 S T S T Put option, X = $400 20 400 – S T 0 Total 420 400 S T Profit = payoff – $420 –20 S T – 420 Sally’s strategy Position Cost Payoff S T 390 S T > 390 Stock index 400 S T S T Put option, X = $390 15 390 – S T 0 Total 415 390 S T Profit = payoff – $415 –25 S T – 415 Profit Joe Sally -20 -25 390 400 S T 20-15
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b. Sally does better when the stock price is high, but worse when the stock price is low. The break-even point occurs at S T = $395, when both positions provide losses of $20. c. Sally’s strategy has greater systematic risk. Profits are more sensitive to the value of the stock index. 24. a., b. (See graph below) This strategy is a bear spread. Initial proceeds = $9 – $3 = $6 The payoff is either negative or zero: Position S T < 50 50 S T 60 S T > 60 Buy call, X = $60 0 0 S T – 60 Write call, X = $50 0 –(S T – 50) –(S T – 50) Total 0 –(S T – 50) –10 c. Breakeven occurs when the payoff offsets the initial proceeds of $6, which occurs at stock price S T = $56. The investor must be bearish: the position does worse when the stock price increases. 0 S T 50 60 6 -10 - 4 Profit Payoff 20-16
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25. Buy a share of stock, write a call with X = $50, write a call with X = $60, and buy a call with X = $110.
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