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Answers to EndofChapter Problems
B367
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
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B368
This implies the percentage increase is if the stock price increases will be 21 percent, and the
percentage decrease if the stock price falls will be 17 percent. The monthly interest rate is:
Monthly interest rate = 0.05/12
Monthly interest rate = 0.0042
Next, we need to find the risk neutral probability of a price increase or decrease, which will be:
0.0042 = 0.21(Probability of rise) –0.17(1 – Probability of rise)
Probability of rise = 0.4643
And the probability of a price decrease is:
Probability of decrease = 1 – 0.4643
Probability of decrease = 0.5357
The following figure shows the stock price and put price for each possible move over the next two
months:
Stock price (D) $ 91.69
Put price $0
Stock price (B) $ 76.00
Put price $ 3.73
Stock price(A) $ 63.00
Stock price (E) $ 63.00
Put price $ 11.21
Put price $ 7.00
Stock price (C) $ 52.22
Put price $ 17.78
Stock price (F) $ 43.29
Put price $ 26.71
The stock price at node (A) is the current stock price. The stock price at node (B) is from an up
move, which means:
Stock price (B) = $63(1.2064)
Stock price (B) = $76.00
And the stock price at node (D) is two up moves, or:
Stock price (D) = $63(1.2064)(1.2064)
Stock price (D) = $91.69
The stock price at node (C) is from a down move, or:
Stock price (C) = $63(0.8289)
Stock price (C) = $52.22
And the stock price at node (F) is two down moves, or:
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 Spring '08
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