# Ross5eChap24sm - 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 25,000 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 \$55. 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 d 1 = [ln(55/55) + {0.054 + ½(0.42 2 )}(4) ] / (0.42 2 *4) 1/2 = 0.677 d 2 = 0.677 – (0.42 2 *4) 1/2 = –0.1628 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(0.677) =0.7518 N(d 2 ) = N(–0.1628) = 0.4365 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 ) C = (55)(0.7518) – (55)e –(0.054)(4) (0.4365) = \$22.005 The Black–Scholes Price of one call option is \$22.005. Since Mr. Levin was granted 25,000 options, the current value of his options package is \$550,133 (= 25,000 * \$22.005). 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 \$550,000, Mr. Levin would prefer to be compensated with the options rather than with the immediate bonus. 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 The total compensation package consists of an annual salary in addition to 10,000 at–the–money stock options. First, we will 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, which will be: PV(Salary) = \$400,000 3 09 . 0 A Answers to End-of-Chapter Problems B-111

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PV(Salary) = \$1,012,517.87 Next, we can use the Black–Scholes model to determine the value of the stock options. Doing so, we find: d 1 = [ln(S/K) + (r + ½ σ 2 )(t) ] / ( σ 2 t) 1/2 d 1 = [ln(\$40/\$40) + (0.05 + 0.68 2 /2) (3)] / (0.68)(3 ) = 0.7163 d2 = 0.7163 – (0.68 )(3) = –0.4615 Find N(d 1 ) and N(d 2 ), the area under the normal curve from negative infinity to d1 and negative infinity to d 2 , respectively. Doing so: N(d 1 ) = N(0.7163) = 0.7631 N(d 2 ) = N(–0.4615) = 0.3222 Now we can find the value of each option, which will be: C = S N(d 1 ) – Ke ––rt N(d 2 ) C = \$40(0.7631) – (\$40e –0.05(3) )(0.3222) C = \$19.43 Since the option grant is for 10,000 options, the value of the grant is: Grant value = 10,000(\$19.43) Grant value = \$194,303.49 The total value of the contract is the sum of the present value of the salary, plus the option value, or: Contract value = \$1,012,517.87 + 4194,303.19 Contract value = \$1,206,821.05 24.3
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## This note was uploaded on 07/18/2010 for the course ECONMICS ECM359 taught by Professor Matazi during the Summer '10 term at University of Toronto- Toronto.

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Ross5eChap24sm - Chapter 24: Options and Corporate Finance:...

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