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Chapter 19
Monte Carlo Valuation
Question 19.1.
The histogram should resemble the uniform density, the mean should be close to 0.5, and the
standard deviation should be close to 1
/
√
12
=
0
.
2887.
Question 19.2.
The histogram should be similar to a standard normal density (“bell” shaped). Since a uniform
distribution has a mean of 0.5 and a variance of 1/12, the mean of
∑
12
i
=
1
u
i
−
6 is zero and the
variance (& standard deviation) will be one since
var
Ã
12
X
i
=
1
u
i
!
=
12
var (u
i
)
=
1.
Question 19.3.
The mean of
e
x
1
should be close to
e
1
/
2
=
1
.
6487 and the mean of
e
x
2
should be close to
e
.
7
+
1
.
5
=
9
.
025.
Question 19.4.
The standard deviation of the estimate will be
s
n
/
√
n
where
s
n
is the sample standard deviation of
the
n
simulations. Since
s
n
is close to 2.9,
n
=
84000 should give a standard error close to 0.01.
Question 19.5.
1
/S
1
=
exp
(
−
.
08
+
.
3
2
/
2
+
.
3
z
)
/
40 generates the simulations. The mean of which should be
close to
e
−
.
035
+
.
3
2
/
2
/
40
=
.
02525. This should also be the forward price.
Question 19.6.
The simulations should be generated by
S
1
=
100 exp
(
.
06
−
.
4
2
/
2
+
.
4
z
)
where
z
is standard
normal. The claim prices should be
e
−
.
06
S
α
where
α
is the relevant power and the
S
α
is the average
from the simulations. These values should be close to
100
α
exp
³
(α
−
1
)
³
.
06
+
α
2
.
4
2
´
.
Using this, the three values should be close to 12461, 9
.
51, and
.
000135 respectively.
249
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View Full DocumentPart 5 Advanced Pricing Theory
Question 19.7.
The fve values should be close to 10366.56, 1.004, 96.95, 10
−
4
, and 1
,
261
,
120 respectively.
Question 19.8.
By log normality
P (S
t
<
95
)
=
P
³
100 exp
³
.
1
−
.
2
2
/
2
´
t
+
.
2
√
tz
´
<
95
´
P
Ã
z<
ln
(
95
/
100
)
−
(
.
1
−
.
2
2
/
2
)
t
.
2
√
t
!
with
t
=
1
/
365 this is
N (
−
4
.
9207
)
=
4
×
10
−
7
. This magnitude negative return should, on av
erage, occur once every 2.5 million days. With
t
=
1
/
252 (i.e. one trading day) this becomes
N (
−
4
.
0965
)
=
2
.
097
×
10
−
5
; making such a drop is similarly unlikely.
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 Spring '06
 Adam

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