The cost of the portfolio is s 2p u 60 2p u the

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Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P u = $60 + 2P u The payoff for the riskless portfolio equals $72: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Buy 2 puts 24 0 Total 72 72 Therefore, find the value of the put by solving: $60 + 2P u = $72/1.06 P u = $3.962 To compute P d , compute the hedge ratio: 0 . 1 32 48 28 12 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 = $40 + P d 21-12
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The payoff for the riskless portfolio equals $60: Riskless Portfolio S = 32 S = 48 Buy 1 share 32 48 Buy 1 put 28 12 Total 60 60 Therefore, find the value of the put by solving: $40 + P d = $60/1.06 P d = $16.604 To compute P, compute the hedge ratio: 6321 . 0 40 60 604 . 16 962 . 3 dS uS P P H 0 0 d u = = = Form a riskless portfolio by buying 0.6321 of a share and buying one put. The cost of the portfolio is: 0.6321S + P = $31.605 + P The payoff for the riskless portfolio equals $41.888: Riskless Portfolio S = 40 S = 60 Buy 0.6321 share 25.284 37.926 Buy 1 put 16.604 3.962 Total 41.888 41.888 Therefore, find the value of the put by solving: $31.605 + P = $41.888/1.06 P = $7.912 d. According to put-call-parity: C = S 0 + P PV(X) = $50 + $7.912 $60/(1.06 2 ) = $4.512 This is the value of the call calculated in part (b) above. 33. 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. 21-13
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34. 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. 35. 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.
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