The cost of the portfolio is s p d 95 p d the payoff

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The cost of the portfolio is: S + P d = $95 + P d 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 21-8
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Chapter 21 - Option Valuation 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. 28. If r = 0, then one should never exercise a put early. There is no “time value cost” to waiting to exercise, but there is a “volatility benefit” from waiting. To show this more rigorously, consider the following portfolio: lend $X and short one share of stock. The cost to establish the portfolio is (X – S 0 ). The payoff at time T (with zero interest earnings on the loan) is (X – S T ). In contrast, a put option has a payoff at time T of (X – S T ) if that value is positive, and zero otherwise. The put’s payoff is at least as large as the portfolio’s, and therefore, the put must cost at least as much as the portfolio to purchase. Hence, P (X – S 0 ), and the put can be sold for more than the proceeds from immediate exercise. We conclude that it doesn’t pay to exercise early. 29. a.Xe - rT b. X c.0 d. 0 e.It is optimal to exercise immediately a put on a stock whose price has fallen to zero. The value of the American put equals the exercise price. Any delay in exercise lowers value by the time value of money. 21-9
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Chapter 21 - Option Valuation 30. Step 1 : Calculate the option values at expiration. The two possible stock prices and the corresponding call values are: uS 0 = 120 C u = 20 dS 0 = 80 C d = 0 Step 2 : Calculate the hedge ratio. 2 1 80 120 0 20 dS uS C C H 0 0 d u = - - = - - = Therefore, form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2C Step 3 : Show that the payoff for the riskless portfolio equals $80: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Write 2 calls 0 -40 Total 80 80 Therefore, find the value of the call by solving: $100 – 2C = $80/1.10 C = $13.636 Notice that we did not use the probabilities of a stock price increase or decrease. These are not needed to value the call option. 31. The two possible stock prices and the corresponding call values are: uS 0 = 130 C u = 30 dS 0 = 70 C d = 0 The hedge ratio is: 2 1 70 130 0 30 dS uS C C H 0 0 d u = - - = - - = Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2C The payoff for the riskless portfolio equals $70: Riskless Portfolio S = 70 S = 130 Buy 1 share 70 130 Write 2 calls 0 -60 Total 70 70 Therefore, find the value of the call by solving: $100 – 2C = $70/1.10 C = $18.182 Here, the value of the call is greater than the value in the lower-volatility scenario.
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