Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
RafałM.Łochowski
(
μ
ˆ
M
,
σ
ˆ
M
,
m
ˆ
M
,
s
ˆ
M
,
p
ˆ
0
,
M
)
=
arg
μ
,
σ
max
,
m
,
s
,
p
0
∏
i
=
n
1
⎛
⎜
⎜
⎜
⎝
∑
j
3
=
0
p
j
exp
2
π
(
-
σ
(
2
r
i
2
(
σ
-
μ
+
2
-
+
j
j
j
⋅
⋅
m
s
⋅
2
)
2
)
s
)
2
⎞
⎟
⎟
⎟
⎠
.
(
μ
ˆ
M
,
Intab.3wepresenttheobtainedmaximumlikelihoodestimates
σ
ˆ
M
,
m
ˆ
M
,
s
ˆ
M
,
p
ˆ
0
,
M
)
fornineassetsconsidered.Theestimationwasbased
onthedatafromtheperiod1stofJanuary2011-20thofSeptember2011.
InthemaximumlikelihoodestimationtheoptimprocedureofRenvironment
wasutilized.
Table3
Maximumlikelihoodestimatorsforthejump-diffusionmodelparametersbased
onthedatafromtheperiod1stofJanuary2011-20thofSeptember2011
ASSECO
GTC
KERNEL
KGHM
ORLEN
PKOBP
TAURON
TPSA
TVN
Asset
-0.0014
-0.0009
-0.0026
-0.0006
-0.0008
-0.0002
0.0003
0.0002
0.0023
μ
ˆ
M
0.0138
0.0213
0.0125
0.0182
0.0150
0.0126
0.0122
0.0153
0.0167
σ
ˆ
M
-0.0023
-0.0361
-0.0060
-0.0034
-0.0085
-0.0060
-0.0264
-0.0088
0.0064
m
ˆ
M
0.0351
0.0329
0.0332
0.0327
0.0387
0.0283
0.0357
0.0398
0.0503
s
ˆ
M
0.8494
0.8984
0.7389
0.8922
0.8208
0.8189
0.9017
0.9085
0.9055
p
ˆ
0
,
M
Thereasonforconsideringtheperiod1stofJanuary2011-20thofSeptem-
ber2011insteadoftheperiod1stofJanuary2011-31stofJuly2011wassuch
thatthemodelsobtainedfromtheshorterperiodfailtofitthedistribution
ofthedailyreturnsduringtheremainingperiodoftheyear2011.Thiswas
testedwiththeKolmogorov-Smirnovgoodnessoffittests.
Remark.Theobtainedestimatesfor
pjustifythesimplificationthat
0
p
k
=
0
for
k
≥
4
.
Indeed,sincefor
Δt
=
1
,
p
0
=
e
-
λ
Δ
t
=
e
-
λ
,
then
λ
=
ln
p
0
-
1
and
p
k
=
p
0
(
ln
p
0
-
1
)
k
/
k
!.
Now,
since
for
p
0
>
e
-
k
the
function
p
0
a
p
k
=
p
0
(
ln
p
0
-
1
)
k
/
k
!
isdecreasing,forallassetswehave
48
