bkmsol_ch21

# Portfolio s 90 s 110 buy 05 invest in t total 100 110

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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

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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 The payoff for the riskless portfolio equals \$60.53: Riskless Portfolio S = 95 S = 110 Buy 0.5344 share 50.768 58.784 Buy 1 put 9.762 1.746 Total 60.530 60.530 Therefore, find the value of the put by solving: \$53.44 + P = \$60.53/1.05 P = \$4.208 Finally, we verify this result using put-call parity. Recall from Example 21.1 that: C = \$4.434 Put-call parity requires that: P = C + PV(X) – S \$4.208 = \$4.434 + (\$110/1.05 2 ) \$100 Except for minor rounding error, put-call parity is satisfied. 21-10
32. a. The two possible values of the index in the first period are: uS 0 = 1.20 × 50 = 60 dS 0 = 0.80 × 50 = 40 The possible values of the index in the second period are: uuS 0 = (1.20) 2 × 50 = 72 udS 0 = 1.20 × 0.80 × 50 = 48 duS 0 = 0.80 × 1.20 × 50 = 48 ddS 0 = (0.80) 2 × 50 = 32 b. The call values in the second period are: C uu = 72 60 = 12 C ud = C du = C dd = 0 Since C ud = C du = 0, then C d = 0. To compute C u , first compute the hedge ratio: 2 1 48 72 0 12 udS uuS C C H 0 0 ud uu = = = Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C u = \$60 – 2C u The payoff for the riskless portfolio equals \$48: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Write 2 calls 0 -24 Total 48 48 Therefore, find the value of the call by solving: \$60 – 2C u = \$48/1.06 C u = \$7.358 To compute C, compute the hedge ratio: 3679 . 0 40 60 0 358 . 7 dS uS C C H 0 0 d u = = = Form a riskless portfolio by buying 0.3679 of a share and writing one call. The cost of the portfolio is: 0.3679S – C = \$18.395 – C 21-11

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The payoff for the riskless portfolio equals \$14.716: Riskless Portfolio S = 40 S = 60 Buy 0.3679 share 14.716 22.074 Write 1 call 0.000 7.358 Total 14.716 14.716 Therefore, find the value of the call by solving: \$18.395 – C = \$14.716/1.06 C = \$4.512 c. The put values in the second period are: P uu = 0 P ud = P du = 60 48 = 12 P dd = 60 32 = 28 To compute P u , first compute the hedge ratio: 2 1 48 72 12 0 udS uuS P P H 0 0 ud uu = = = Form a riskless portfolio by buying one share of stock and buying two puts.
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