Answers to End-of-Chapter Problems
B-367
Chapter 24: Options and Corporate Finance:
Extensions and Applications
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
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
24.4
Using the binomial mode, we will find the value of u and d, which are:
u = e
σ
/
√
t
u = e
.65/
√
12
u = 1.21
d = 1 / u
d = 1 / 1.21
d = 0.83