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# American options give the investor greater

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American options give the investor greater flexibility than European options since the investor can choose whether to exercise early. When the stock pays a dividend, the option to exercise a call early can be valuable. But regardless of the dividend, a European option (put or call) never sells for more than an otherwise-identical American option. b. C = S 0 + P PV(X) = \$43 + \$4 \$45/1.055 = \$4.346 Note: we assume that Abaco does not pay any dividends. c. An increase in short-term interest rate PV(exercise price) is lower, and call value increases. An increase in stock price volatility the call value increases. A decrease in time to option expiration the call value decreases. 30. a. uS 0 = 110 P u = 0 dS 0 = 90 P d = 10 The hedge ratio is: 2 1 90 110 10 0 dS uS P P H 0 0 d u = = = A portfolio comprised of one share and two puts provides a guaranteed payoff of \$110, with present value: \$110/1.05 = \$104.76 Therefore: S + 2P = \$104.76 \$100 + 2P = \$104.76 P = \$2.38 b. Cost of protective put portfolio = \$100 + \$2.38 = \$102.38 21-8

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c. Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is –0.5, the portfolio consists of (1 – 0.5) = 0.5 shares of stock, which costs \$50, and the remaining funds (\$52.38) invested in T-bills, earning 5% interest. Portfolio S = 90 S = 110 Buy 0.5 shares 45 55 Invest in T-bills 55 55 Total 100 110 This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put. 31. The put values in the second period are: P uu = 0 P ud = P du = 110 104.50 = 5.50 P dd = 110 90.25 = 19.75 To compute P u , first compute the hedge ratio: 3 1 50 . 104 121 50 . 5 0 udS uuS P P H 0 0 ud uu = = = Form a riskless portfolio by buying one share of stock and buying three puts. The cost of the portfolio is: S + 3P u = \$110 + 3P u The payoff for the riskless portfolio equals \$121: Riskless Portfolio S = 104.50 S = 121 Buy 1 share 104.50 121.00 Buy 3 puts 16.50 0.00 Total 121.00 121.00 Therefore, find the value of the put by solving: \$110 + 3P u = \$121/1.05 P u = \$1.746 To compute P d , compute the hedge ratio: 0 . 1 25 . 90 50 . 104 75 . 19 50 . 5 ddS duS P P H 0 0 dd du = = = Form a riskless portfolio by buying one share and buying one put. The cost of the portfolio is: S + P d = \$95 + P d 21-9
The payoff for the riskless portfolio equals \$110: Riskless Portfolio S = 90.25 S = 104.50 Buy 1 share 90.25 104.50 Buy 1 put 19.75 5.50 Total 110.00 110.00 Therefore, find the value of the put by solving: \$95 + P d = \$110/1.05 P d = \$9.762 To compute P, compute the hedge ratio: 5344 . 0 95 110 762 . 9 746 . 1 dS uS P P H 0 0 d u = = = Form a riskless portfolio by buying 0.5344 of a share and buying one put. The cost of the portfolio is: 0.5344S + P = \$53.44 + P

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American options give the investor greater flexibility than...

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