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

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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 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 21-16
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Chapter 21 - Option Valuation 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. 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 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 - = - - = - - = 21-17
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Chapter 21 - Option Valuation 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. 5. a.(i) Index increases to 1402. The combined portfolio will suffer a loss. The written calls expire in the money; the protective put purchased expires worthless. Let’s analyze the outcome on a per-share basis. The payout for each call option is $52, for a total cash outflow of $104. The stock is worth $1,402. The portfolio will thus be worth: $1,402 - $104 = $1,298 The net cost of the portfolio when the option positions are established is: $1,336 + $16.10 (put) - [2 × $8.60] (calls written) = $1,334.90 (ii) Index remains at 1336. Both options expire out of the money. The portfolio will thus be worth $1,336 (per share), compared to an initial cost 30 days earlier of $1,334.90. The portfolio experiences a very small gain of $1.10. (iii) Index declines to 1270. The calls expire worthless. The portfolio will be worth $1,330, the exercise price of the protective put. This represents a very small loss of $4.90 compared to the initial cost 30 days earlier of $1,334.90 b. (i) Index increases to 1402. The delta of the call approaches 1.0 as the stock goes deep into the money, while expiration of the call approaches and exercise becomes essentially certain. The put delta approaches zero.
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