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CHAPTER 21
Valuing Options
Answers to Practice Questions
1. a.
188
844
.
0
/
1
;
185
.
1
5
.
0
24
.
0
=
=
=
=
u
d
e
u
$45
$53.33
$37.98
$63.19
$45.01
$45.01
$32.06
887
.
0
/
1
;
127
.
1
25
.
0
24
.
0
=
=
=
=
u
d
e
u
$45
$50.72
$57.16
$64.41
$35.41
$50.70
$39.92
$44.98
$39.91
$72.60
$57.14
$57.14
$44.97
$44.97
$35.40
$31.41
$35.40
$27.86

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*Sign up* b.
189
809
.
0
/
1
,
236
.
1
5
.
0
3
.
0
=
=
=
=
u
d
e
u
$45
$55.62
$36.41
$68.75
$45.00
$45.00
$29.46
$45
$52.29
$60.76
$70.60
$33.36
$52.32
$38.75
$45.02
$38.77
$82.04
$60.79
$60.79
$45.05
$45.05
$33.38
$28.72
$33.38
$24.73
861
.
0
/
1
;
162
.
1
25
.
0
3
.
0
=
=
=
=
u
d
e
u

a.
Let p equal the probability of a rise in the stock price.
Then, if investors are risk-
neutral:
(p
×
0.15) + (1 - p)
×
(-0.13) = 0.10
p = 0.821
The possible stock prices next period are:
$60
×
1.15 = $69.00
$60
×
0.87 = $52.20
Let X equal the break-even exercise price.
Then the following must be
true:
X – 60 = (p)($0) + [(1 – p)(X – 52.20)]/1.10
That is, the value of the put if exercised immediately equals the value of
the put if it is held to next period.
Solving for X, we find that the break-
even exercise price is $61.52.
b.
If the interest rate is increased, the value of the put option decreases.
If there is an
increase in:
The change in the put
option price is:
Stock price (P)
Negative
Exercise price (EX)
Positive
Interest rate (r
f
)
Negative
Time to expiration (t)
Positive
Volatility of stock price (
σ
)
Positive
Consider the following base case assumptions:
P = 100, EX = 100, r
f
= 5%, t = 1,
σ
= 50%
Then, using the Black-Scholes model, the value of the put is $16.98
The base case value along with values computed for various changes in the
assumed values of the variables are shown in the table below:
Black-Scholes
put value:
Base case
16.98
P = 120
11.04
EX = 120
29.03
r
f
= 10%
14.63
t = 2
21.94
σ
= 100%
35.04
190

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*Sign up* a.
The future stock prices of Matterhorn Mining are:
With dividend
Ex-dividend
Let p equal the probability of a rise in the stock price.
Then, if investors
are risk-neutral:
(p
×
0.25) + (1 - p)
×
(-0.20) = 0.10
p = 0.67
Now, calculate the expected value of the call in month 6.
If stock price decreases to SFr80 in month 6, then the call is worthless.
If
stock price increases to SFr125, then, if it is exercised at that time, it has a
value of (125 – 80) = SFr45.
If the call is not exercised, then its value is:
Therefore, it is preferable to exercise the call.
The value of the call in month 0 is:
b.
The future stock prices of Matterhorn Mining are:
With dividend
Ex-dividend
191
100
80
125
60
105
75
48
131.25
84
?
0
0
0
32.42
51.25
4
SFr27.41
1.10
0)
(0.33
5)
(0.67
4
=
×
+
×
100
80
125
51.2
80
125
64
100
SFr32.42
1.10
4)
(0.33
51.25)
(0.67
=
×
+
×

Let p equal the probability of a rise in the price of the stock.
Then, if
investors are risk-neutral:
(p
×
0.25) + (1 - p)
×
(-0.20) = 0.10
p = 0.67
Now, calculate the expected value of the call in month 6.

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