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THEGEOMETRICBROWNIANMOTIONMODEL…
Table2
Normalitytestsfordailylogarithmicreturns
intheperiod1stJanuary2011-20thDecember2011
ASSECO
GTC
KERNEL
KGHM
ORLEN
PKOBP
TAURON
TPSA
TVN
Asset
Shapiro-Wilktest
statistic'sp-value
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.0003
0.0002
statistic'sp-value
Jarque-Beratest
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.0048
Thusweseethatempiricaltestsshowthatthenormalityofreturnsdoesnot
holdfordailyreturnsintheanalyzedperiodsformajorityoftheassets.
Thisassumptionholdsapproximatelyforlongertimeperiods
Δ(e.g.
t
Δt
≥
3
months),butontheotherhand,theassumptionaboutconstantvalueofvolati-
lityanddrifinlongerperiodsisunrealistic.
Therejectionofthehypothesisofnormalityofdailylogarithmicreturns,
whenthesamplefromwholeyear2011isconsidered,hasitsrootsinthehigh
marketvolatilityinthesecondhalfoftheyear2011.Infact,inJuneandthe
followingmonthsof2011,financialmarketsreactedstronglytothesignsofthe
deepeningEurozonecrisis.Forexample,inJune2011StandardandPoor's
downgradedGreece'sdebtratingtothelowestintheworld(CCC),following
thefindingsofabilateralEU-IMFaudit.On6July2011theratingsagency
Moody'shadcutPortugal'screditratingtojunkstatus,Moody'salsolaunched
speculationthatPortugalmayfollowGreeceinrequestingasecondbailout.
2.Thejump-diffusionmodelvs.thegeometricBrownian
motionmodel
2.1.Thejump-diffusionmodel
DuetothelimitationsofthegeometricBrownianmotionmodelinmode-
lingthedistributionoflogarithmicstockreturnsinshorterperiods,RobertMer-
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