Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
THEGEOMETRICBROWNIANMOTIONMODEL…
p
ˆ
k
≤
⎡
⎢
⎣
p
0
(
ln
p
0
-
1
)
4
/
4
!
⎤
⎥
⎦
p
0
=
0
.
7389
<
0
.
0003
for
k
≥
4
.
Similarly,weconsideredthemaximumlikelihoodestimatorsof
μ
,
σ
inthegeometricBrownianmotionmodel.Inthiscaseitispossibletogive
explicitformulasfortheseestimators,denotedrespectivelyby
μ
ˆ
BS
,
σ
ˆ
BS
.
Theformulasreadas
μ
ˆ
BS
=
r
=
1
n
∑
i
=
n
1
r
i
,
σ
ˆ
BS
=
1
n
∑
i
n
=
1
(
r
i
-
r
)
2
and,usingthedatafromtheperiod1stofJanuary2011-20thofSeptember
2011,weobtainestimatorspresentedintab.4.
Table4
MaximumlikelihoodestimatorsforthegeometricBrownianmotionmodelparametersbased
onthedatafromtheperiod1stofJanuary2011-20thofSeptember2011
ASSECO
GTC
KERNEL
KGHM
ORLEN
PKOBP
TAURON
TPSA
TVN
Asset
-0.0006
-0.0004
-0.0018
-0.0049
-0.0014
-0.0017
-0.0015
-0.0002
-0.0011
μ
ˆ
BS
0.0200
0.0266
0.0229
0.0221
0.0231
0.0186
0.0169
0.0210
0.0234
σ
ˆ
BS
Duetobiggernumberofparameters,theobtainedjump-diffusionmodel
fitsbetterthanthegeometricBrownianmotionmodelthedistribution
ofthedailyreturnsintheperiod1stofJanuary2011-20thofSeptember2011.
Thismaybemeasurede.g.withtheKolmogorov-Smirnovstatistics
D
M
=
sup
x
E
R
1
n
∑
i
=
n
1
I
{
x
≥
r
i
}
-
j
∑
3
=
0
p
ˆ
j
,
M
Φ
⎛
⎜
⎜
⎝
x
-
σ
ˆ
μ
ˆ
M
M
+
-
j
j
⋅
⋅
s
ˆ
m
M
ˆ
2
M
⎞
⎟
⎟
⎠
D
BS
=
sup
x
E
R
1
n
∑
i
n
=
1
I
{
x
≥
r
i
}
-
Φ
⎛-
⎜
⎜
⎝
x
σ
ˆ
BS
μ
ˆ
BS
⎞
⎟
⎟
⎠
49
