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Chapter 23: Options and Corporate Finance:
Extensions and Applications
23.1
a.
The inputs to the BlackScholes 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 continuouslycompounded riskfree 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 atthemoney, the strike price of each option is equal to
the current value of one share, or $50.
Therefore, the BlackScholes 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 BlackScholes formula, the price of a European call option (C) on a nondividend
paying common stock is:
C
= SN(d
1
) – Ke
rt
N(d
2
)
= (50)(0.7704) – (50)e
(0.06)(4)
(0.3974)
= $22.89
The BlackScholes Price of one call option is $22.89.
Since Mr. Levin was granted 20,000 options, the current value of his options package is $457,794
(= 20,000 * $22.89).
b.
Because Mr. Levin is riskneutral, 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.
c.
If Mr. Levin is riskaverse, 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.
B143
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View Full Document23.2
Jared’s total compensation package consists of an annual salary of $500,000 for 3 years in addition to
10,000 atthemoney 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 threeyear annuity, discounted at 10%.
PV(Annual Salary Payments) = $500,000 A
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 Spring '08
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