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