The less volatile the underlying stock price the less

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stock price decreases. The less volatile the underlying stock price, the less the chance of extreme price movements and the lower the probability that the option expires in the money. This makes the participation feature on the upside less valuable. The value of the call option is expected to increase if the time to expiration of the option increases. The longer the time to expiration, the greater the chance that the option will expire in the money resulting in an increase in the time premium component of the option’s value. 21-14
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Chapter 21 - Option Valuation b. i. When European options are out of the money, investors are essentially saying that they are willing to pay a premium for the right, but not the obligation, to buy or sell the underlying asset. The out-of-the-money option has no intrinsic value, but, since options require little capital (just the premium paid) to obtain a 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). 3. 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. i) An increase in short-term interest rate PV(exercise price) is lower, and call value increases. ii) An increase in stock price volatility the call value increases. iii) A decrease in time to option expiration the call value decreases. 4. a.The two possible values of the index in the first period are: uS 0 = 1.20 × 50 = 60 dS 0 = 0.80 × 50 = 40 The possible values of the index in the second period are: uuS 0 = (1.20) 2 × 50 = 72 udS 0 = 1.20 × 0.80 × 50 = 48 duS 0 = 0.80 × 1.20 × 50 = 48 ddS 0 = (0.80) 2 × 50 = 32 21-15
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Chapter 21 - Option Valuation b. The call values in the second period are: C uu = 72 60 = 12 C ud = C du = C dd = 0 Since C ud = C du = 0, then C d = 0. To compute C u , first compute the hedge ratio: 2 1 48 72 0 12 udS uuS C C H 0 0 ud uu = - - = - - = Form a riskless portfolio by buying one share of stock and writing two calls.
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