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# 37 the two possible stock prices and the

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37. The two possible stock prices and the corresponding put values are: uS 0 = 120 P u = 0 dS 0 = 80 P d = 20 The hedge ratio is: 2 1 80 120 20 0 dS uS P P H 0 0 d u = = = Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P = 100 + 2P The payoff for the riskless portfolio equals \$120: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Buy 2 puts 40 0 Total 120 120 Therefore, find the value of the put by solving: \$100 + 2P = \$120/1.10 P = \$4.545 According to put-call parity: P + S = C + PV(X) Our estimates of option value satisfy this relationship: \$4.545 + \$100 = \$13.636 + \$100/1.10 = \$104.545 21-15

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If we assume that the only possible exercise date is just prior to the ex-dividend date, then the relevant parameters for the Black-Scholes formula are: S 0 = 60 r = 0.5% per month X = 55 σ = 7% T = 2 months In this case: C = \$6.04 If instead, one commits to foregoing early exercise, then we reduce the stock price by the present value of the dividends. Therefore, we use the following parameters: S 0 = 60 – 2e (0.005 × 2) = 58.02 r = 0.5% per month X = 55 σ = 7% T = 3 months In this case, C = \$5.05 The pseudo-American option value is the higher of these two values: \$6.04 39. a. (i) Index increases to 701. 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 \$26, for a total cash outflow of \$52. The stock is worth \$701. The portfolio will thus be worth: \$701 \$52 = \$649 The net cost of the portfolio when the option positions are established is: \$668 + \$8.05 (put) [2 × \$4.30] (calls written) = \$667.45 (ii) Index remains at 668. Both options expire out of the money. The portfolio will thus be worth \$668 (per share), compared to an initial cost 30 days earlier of \$667.45. The portfolio experiences a very small gain of \$0.55. (iii) Index declines to 635. The calls expire worthless. The portfolio will be worth \$665, the exercise price of the protective put. This represents a very small loss of \$2.45 compared to the initial cost 30 days earlier of \$667.45. b. (i) Index increases to 701. 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. (ii) Index remains at 668. Both options expire out of the money. Delta of each approaches zero as expiration approaches and it becomes certain that the options will not be exercised. (iii) Index declines to 635. The call is out of the money as expiration approaches. Delta approaches zero. Conversely, the delta of the put approaches 1.0 as exercise becomes certain. 21-16
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37 The two possible stock prices and the corresponding put...

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