chprobSM_ch24 - Chapter 24 Options and Corporate Finance...

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Chapter 24: Options and Corporate Finance: Extensions and Applications 24.1 a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset ( σ 2 ), and the continuously-compounded risk-free interest rate (r). Mr. Levin has been granted 20,000 European call options on Mountainbrook’s stock with 4 years until expiration. Since these options were granted at-the-money, the strike price of each option is equal to the current value of one share, or $50. Therefore, the Black-Scholes inputs are: S = $50 σ 2 = 0.25 K = $50 r = 0.06 t = 4 After identifying the inputs, solve for d 1 and d 2 : d 1 = [ln(S/K) + (r + ½ σ 2 )(t) ] / ( σ 2 t) 1/2 = [ln(50/50) + {0.06 + ½(0.25)}(4) ] / (0.25*4) 1/2 = 0.7400 d 2 = d 1 - ( σ 2 t) 1/2 = 0.7400 - (0.25*4) 1/2 = -0.2600 Find N(d 1 ) and N(d 2 ), the area under the normal curve from negative infinity to d 1 and negative infinity to d 2 , respectively. N(d 1 ) = N(0.7400) = 0.7704 N(d 2 ) = N(-0.2600) = 0.3974 According to the Black-Scholes formula, the price of a European call option (C) on a non- dividend paying common stock is: C = SN(d 1 ) – Ke -rt N(d 2 ) = (50)(0.7704) – (50)e -(0.06)(4) (0.3974) = $ 22.8897 The Black-Scholes Price of one call option is $ 22.8897. Since Mr. Levin was granted 20,000 options, the current value of his options package is $457,794 (= 20,000 * $ 22.8897). b. Because Mr. Levin is risk-neutral, you should recommend the alternative with the highest net present value. Since the expected value of the stock option package is worth more than $450,000, Mr. Levin would prefer to be compensated with the options rather than with the immediate bonus. Answers to End-of-Chapter Problems B-111
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c. If Mr. Levin is risk-averse, he may or may not prefer the stock option package to the immediate bonus. Even though the stock option package has a higher net present value, he may not prefer it because it is undiversified. The fact that he cannot sell his options prematurely makes it much more risky than the immediate bonus. Therefore, we cannot say which alternative he would prefer. 24.2 Kimberleigh’s total compensation package consists of an annual salary of $500,000 for 3 years in addition to 10,000 at-the-money stock options. First, find the present value of the salary payments. Since the payments occur at the end of the year, the payments can be valued as a three-year annuity, discounted at 10%.
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