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# 11 a straddle is a call and a put the black scholes

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11. A straddle is a call and a put. The Black-Scholes value would be: C + P = S 0 N(d 1 ) Xe –rT N(d 2 ) + Xe –rT [1 N(d 2 )] S 0 [1 N(d 1 )] = S 0 [2N(d 1 ) 1] + Xe –rT [1 2N(d 2 )] On the Excel spreadsheet (Spreadsheet 21.1), the valuation formula would be: B5*(2*E4 1) + B6*EXP( B4*B3)*(1 2*E5) 12. A \$1 increase in a call option’s exercise price would lead to a decrease in the option’s value of less than \$1. The change in the call price would equal \$1 only if: (i) there were a 100% probability that the call would be exercised, and (ii) the interest rate were zero. 21-3

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13. Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore, the value of the put option increases as beta increases. 14. Holding beta constant, the stock with a lot of firm-specific risk has higher total volatility. The option on the stock with higher firm-specific risk is worth more. 15. A call option with a high exercise price has a lower hedge ratio. This call option is less in the money. Both d 1 and N(d 1 ) are lower when X is higher. 16. The rate of return of a call option on a long-term Treasury bond should be more sensitive to changes in interest rates than is the rate of return of the underlying bond. The option elasticity exceeds 1.0. In other words, the option is effectively a levered investment and the rate of return on the option is more sensitive to interest rate swings. 17. Implied volatility has increased. If not, the call price would have fallen as a result of the decrease in stock price. 18. Implied volatility has increased. If not, the put price would have fallen as a result of the decreased time to maturity. 19. The hedge ratio approaches one. As S increases, the probability of exercise approaches 1.0. N(d 1 ) approaches 1.0. 20. The hedge ratio approaches –1.0. As S decreases, the probability of exercise approaches 1. [N(d 1 ) –1] approaches –1 as N(d 1 ) approaches 0. 21. A straddle is a call and a put. The hedge ratio of the straddle is the sum of the hedge ratios of the individual options: 0.4 + (–0.6) = –0.2 22. a. The spreadsheet appears as follows: INPUTS OUTPUTS Standard deviation (annual) 0.3213 d1 0.0089 Maturity (in years) 0.5 d2 -0.2183 Risk-free rate (annual) 0.05 N(d1) 0.5036 Stock Price 100 N(d2) 0.4136 Exercise price 105 B/S call value 8.0000 Dividend yield (annual) 0 B/S put value 10.4076 The standard deviation is: 0.3213 21-4
b. The spreadsheet below shows the standard deviation has increased to: 0.3568 INPUTS OUTPUTS Standard deviation (annual) 0.3568 d1 0.0318 Maturity (in years) 0.5 d2 -0.2204 Risk-free rate (annual) 0.05 N(d1) 0.5127 Stock Price 100 N(d2) 0.4128 Exercise price 105 B/S call value 9.0000 Dividend yield (annual) 0 B/S put value 11.4075 Implied volatility has increased because the value of an option increases with greater volatility. c.

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11 A straddle is a call and a put The Black Scholes value...

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