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THEGEOMETRICBROWNIANMOTIONMODEL…
Letuscommentontheorganizationofthepaper.Inthenextsection
weshortlypresentthegeometricBrownianmotionmodelandtestthenormality
ofdailylogarithmicreturnsofthecompaniesconsidered.Inthethirdsection
wepresentshortlythejump-diffusionmodelandestimatecoefficientsoftwo
modelsconsidered.Next,wepresentthecomparisonofthegoodnessoffittests
ofthemodelsobtained,testedwithanew,independentsample.Wecompare
alsothevaluesof
VaR
95
%
obtainedfromtwomodels.Inthelastsection
wepresentconcludingremarks.
Inallcalculationsweuseddatafromthebossa.plwebpageandRsoftware
forstatisticalcomputingandgraphics.
1.ThegeometricBrownianmotionmodel
anditsconsequences.Normalitytestsforreturns
1.1.ThegeometricBrownianmotionmodel
Theclassicalmodeloftheevolutionofstockprices
Sincontinuoustime
t
t
≥
0
isthegeometricBrownianmotionmodel(cf.[2]).Inthismodelthe
evolutionof
Sisdescribedbythefollowingstochasticdifferentialequation
t
[2]:
dSt=vStdt+
σ
StdBt
where
BisastandardBrownianmotionand
t
ν
,
σ
areconstants,calleddrift
andvolatilityrespectively.Equation[2],given
S
0
,
hasuniquestrongsolution
whichisageometricBrownianmotionprocessgivenbythefollowingformula:
S
t
=
S
0
e
(
ν
-
σ
2
/
2
)
t
+
σ
B
t
=
S
0
e
μ
t
+
σ
B
t
where
μ
=
ν
-
σ
2
2
.
TheimmediateconsequenceofthegeometricBrownianmotionmodel
isthelognormaldistributionofstockpricesatafixedtimeinthefuture,aswell
asthenormalityoflogarithmicreturns.Indeed,wehave:
ln
S
S
t
+
t
Δ
t
=
ln
S
0
e
S
μ
0
(
e
t
+
μ
Δ
t
t
+
)
σ
+
σ
B
t
B
t
+
Δ
t
=
μ
(
t
+
Δ
t
)
+
σ
B
t
+
Δ
t
-
μ
t
-
σ
B
t
=
μ
Δ
t
+
σ
(
B
t
+
Δ
t
-
B
t
)
∼
N
(
μ
Δ
t
,
σ
2
Δ
t
)
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