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1、Temperedstabledistributionsandprocessesinfinance:numericalanalysisMicheleLeonardoBianchi∗,SvetlozarT.Rachev,YoungShinKim,andFrankJ.FabozziAbstract.Mostoftheimportantmodelsinfinancerestontheassumptionthatrandomnessisexplainedthroughanormalrandomvariable.Howeverthereisampleempirica
2、levidenceagainstthenormalityassumption,sincestockreturnsareheavy-tailed,leptokurticandskewed.Partlyinresponsetothoseempiricalinconsistenciesrelativetothepropertiesofthenormaldistribution,asuitablealternativedistributionisthefamilyoftemperedstabledistributions.Ingeneral,theuseof
3、infinitelydivisibledistributionsisobstructedthedifficultyofcalibratingandsimulatingthem.Inthispaper,weaddresssomenumericalissuesresultingfromtemperedstablemodelling,withaviewtowardthedensityapproximationandsimulation.Keywords:stabledistribution,temperedstabledistributions,MonteCa
4、rlo1IntroductionSinceMandelbrotintroducedtheα-stabledistributioninmodellingfinancialassetreturns,numerousempiricalstudieshavebeendoneinbothnaturalandeconomicsciences.TheworksofRachevandMittnik[19]andRachevetal.[18](seealsoreferencestherein),havefocusedattentiononageneralframewor
5、kformarketandcreditriskmanagement,optionpricing,andportfolioselectionbasedontheα-stabledistribution.Whiletheempiricalevidencedoesnotsupportthenormaldistribution,itisalsonotalwaysconsistentwiththeα-stabledistributionalhypothesis.Assetreturnstimeseriespresentheaviertailsrelativet
6、othenormaldistributionandthinnertailsthantheα-stabledistribution.Moreover,thestablescalingpropertiesmaycauseproblemsincalibratingthemodeltorealdata.Anyway,thereisawideconsensustoassumethepresenceofaleptokurticandskewedpatterninstockreturns,asshowedbytheα-stablemodelling.Partlyi
7、nresponsetotheaboveempiricalinconsistencies,andtomaintainsuitablepropertiesofthestablemodel,aproperalternativetotheα-stabledistributionisthefamilyoftemperedstabledistributions.Temperedstabledistributionsmayhaveallmomentsfiniteandexponentialmo-mentsofsomeorder.Thelatterpropertyis
8、essentialintheconstructionoftempered∗Theviewsexpressed
