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Relatively large potential payoff investors are

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relatively large potential payoff, investors are willing to pay that premium even if the option may expire worthless. The Black-Scholes model does not reflect investors’ demand for any premium above the time value of the option. Hence, if investors are willing to pay a premium for an out-of-the-money option above its time value, the Black-Scholes model does not value that excess premium. ii. With American options, investors have the right, but not the obligation, to exercise the option prior to expiration, even if they exercise for non-economic reasons. This increased flexibility associated with American options has some value but is not considered in the Black-Scholes model because the model only values options to their expiration date (European options). 28. S = 100; current value of portfolio X = 100; floor promised to clients (0% return) σ = 0.25; volatility r = 0.05; risk-free rate T = 4 years; horizon of program a. Using the Black-Scholes formula, we find that: d 1 = 0.65, N(d 1 ) = 0.7422, d 2 = 0.15, N(d 2 ) = 0.5596 Put value = $10.27 Therefore, total funds to be managed equals $110.27 million: $100 million portfolio value plus the $10.27 million fee for the insurance program. The put delta is: N(d 1 ) – 1 = 0.7422 – 1 = –0.2578 Therefore, sell off 25.78% of the equity portfolio, placing the remaining funds in T-bills. The amount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is: $110.27 million – $74.22 million = $36.05 million 21-7
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b. At the new portfolio value, the put delta becomes: –0.2779 This means that you must reduce the delta of the portfolio by: 0.2779 – 0.2578 = 0.0201 You should sell an additional 2.01% of the equity position and use the proceeds to buy T-bills. Since the stock price is now at only 97% of its original value, you need to sell: $97 million × 0.0201 = $1.950 million of stock 29. a. American options should cost more (have a higher premium). American options give the investor greater flexibility than European options since the investor can choose whether to exercise early. When the stock pays a dividend, the option to exercise a call early can be valuable. But regardless of the dividend, a European option (put or call) never sells for more than an otherwise-identical American option. b. C = S 0 + P PV(X) = $43 + $4 $45/1.055 = $4.346 Note: we assume that Abaco does not pay any dividends. c. An increase in short-term interest rate PV(exercise price) is lower, and call value increases. An increase in stock price volatility the call value increases. A decrease in time to option expiration the call value decreases. 30. a. uS 0 = 110 P u = 0 dS 0 = 90 P d = 10 The hedge ratio is: 2 1 90 110 10 0 dS uS P P H 0 0 d u = = = A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110/1.05 = $104.76 Therefore: S + 2P = $104.76 $100 + 2P = $104.76 P = $2.38 b. Cost of protective put portfolio = $100 + $2.38 = $102.38 21-8
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c. Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is –0.5, the portfolio consists of (1 – 0.5) = 0.5 shares of stock, which costs $50, and the remaining funds ($52.38) invested in T-bills, earning 5% interest.
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