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Closed-formtransformations.pdf

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CLOSED-FORMTRANSFORMATIONSFROMRISK-NEUTRAL TOREAL-WORLDDISTRIBUTIONS XiaoquanLiu MarkBShackleton StephenJTaylor and XinzhongXu DiscussionPaperNo.04-05 ¥ Closed-formTransformationsfromRisk-neutralto Real-worldDistributions XiaoquanLiu*,MarkBShackleton*, StephenJTaylor*andXinzhongXu** *DepartmentofAccounting&Finance,LancasterUniversity **GuanghuaSchoolofManagement,PekingUniversity December2002 RevisedDecember2003 Contactinformationforcorrespondence: StephenJTaylor,DepartmentofAccounting&Finance,ManagementSchool, LancasterUniversity,EnglandLA14YX,telephone+441524593624,e-mail S.Taylor@lancaster.ac.uk 1 Closed-formTransformationsfromRisk-neutralto Real-worldDistributions Abstract Risk-neutral(RN)andreal-world(RW)densitiesarederivedfromoptionpricesand riskassumptions,andarecomparedwithhistoricaldensitiesobtainedfromtime series.TwoparametricmethodsthatadjustfromRNtoRWdensitiesaredeveloped, firstlyaCRRAriskaversiontransformationandsecondlyastatisticalcalibration. Bothrisktransformationsareestimatedusinglikelihoodtechniques,fortwoflexible buttractableparametricdensityfamilies.ResultsfortheFTSE-100indexshowthat parametricdensitiesderivedfromoptionpriceshavemoreexplanatorypowerthan historicaldensities.Theparametricdensitiesalsohavehigherlikelihoodsthannon- parametricdensitiesestimatedbysplinemethods.Furthermore,thepricingkernel betweenRN&historicaldensitiesisonlyincompatiblewithariskaverse representativeagentwhensplinemethodsprovidetheRNdensities. 2 Closed-formTransformationsfromRisk-neutralto Real-worldDistributions 1.Introduction Optionspricesprovidearichsourceofinformationforestimatingrisk-neutral densities(RNDs)becauseacompletesetofstrikescanbeusedtoinferthe distributionoftheunderlyingassetpricewhentheoptionsexpire.Asaresultmany densityspecificationshavebeenestimatedfromoptionsprices.Parametric specificationsincludeamixtureoflognormals[Ritchey(1990),MelickandThomas (1997)]