Nonparametric Volatility Density Estimation

NonparametriVolatilityDensityEstimationBertvanEs,PeterSpreijKorteweg-deVriesInstituteforMathematisUniversityofAmsterdamPlantageMuidergraht241018TVAmsterdamTheNetherlandsHarryvanZantenCentrumvoorWiskundeenInformatiaKruislaan4131098SJAmsterdamTheNetherlandsJuly19,2001AbstratWeonsidertwokindsofstohastivolatilitymodels.Bothkindsofmodelsontainastationaryvolatilityproess,thedensityofwhih,ataxedinstantintime,weaimtoestimate.Wedisussdisretetimemodelswhereforinstanealogpriepro-essismodeledastheprodutofavolatilityproessandi.i.d.noise.Wealsoonsidersamplesofertainontinuoustimediusionproesses.Thesampledtimeinstantswillbeequidistantwithvanishingdistane.AFouriertypedeonvolutionkerneldensityestimatorbasedonthelogarithmofthesquaredproessesisproposedtoestimatethevolatil-itydensity.Expansionsofthebiasandboundsonthevarianesarederived.Keywords:stohastivolatilitymodels,densityestimation,kerneles-timator,deonvolution,mixingAMSsubjetlassiation:62G07,62M07,62P2011IntrodutionLetSdenotethelogprieproessofsomestokinananialmarket.ItisoftenassumedthatSanbemodelledasthesolutionofastohastidierentialequationor,moregeneral,asanIt^odiusionproess.SoweassumethatweanwritedSt=btdt+tdWt;S0=0;(1.1)or,inintegralform,St=Zt0bsds+Zt0sdWs;(1.2)whereWisastandardBrownianmotionandtheproessesbandareassumedtosatisfyertainregularityonditions(seeKaratzasandShreve(1991))tohavetheintegralsin(1.2)welldened.Inthenanialontext,theproessisalledthevolatilityproess.Inthispaperwemodelasastritlystationarypositiveproesssat-isfyingamixingondition,forexampleanergodidiusionon[0;1)andwemaketheassumptionthatisindependentofW.WewillassumethattheonedimensionalmarginaldistributionofhasadensitywithrespettotheLebesguemeasureon(0;1).Thisistypiallythease,sineinallstohastivolatilitymodelsthatareproposedintheliterature,theevolutionofismodelledbyastohastidierentialequation,mostlyintermsof2.Itisthepurposeofthepapertoestimatethe(marginal)invariantdensityofthevolatilityproess.Themethodthatwewillusefoussesontheestimationofthemarginaldensityoflog2t.Thedensityoftanthenbeobtainedbyasimpletransformationbyusingtheonventionthatisthesquarerootof2.Examplesofsuhvolatilitymodels,allofthemofdiusiontype,aregivenbelow.Theb1;b2andÆarerealonstantsandBisastandardBrow-nianmotion(independentofW).dlog2t=(b1b2log2t)dt+ÆdBt(1.3)d2t=(b1b22t)dt+ÆtdBt(1.4)d2t=(b1b22t)dt+Æ2tdBt(1.5)Inequation(1.3)anOrnstein-Uhlenbekproessisusedtomodeltheloga-rithmofthevolatilityandwasproposedasamodelbyWiggins(1987).Themodelofequation(1.4)wassuggestedbyHeston(1993).ItisthesameastheonethatCox,IngersollandRoss(1985)usedforthetermstrutureofinterestrates.Finally,equation(1.5)arisesinanaturalwayasalimitofaGARCH(1,1)proess,seebelowforanexplanationofthisterminology.Intheexamplesabovetheonditionstoensureanergodistationarysolutionaresatisedbyallofthemforproperhoiesoftheparameters.2Moreover,inalltheseasesweanharaterizetheinvariantdistribu-tion.Underappropriateonditions,seee.g.GihmanandSkorohod(1972)orSkorokhod(1989),forstohastidierentialequationsofthetypedXt=b(Xt)dt+a(Xt)dBtwhereXttakesitsvaluesinanopen(boundedorunbounded)interval(l;r)theinvariantdensityisuptoamultipliativeonstantequaltox7!1a2(x)exp2Zxx0b(y)a2(y)dy;(1.6)wherex0isanarbitraryelementofthestatespae(l;r).Usingthisfor-mulaandsimplyfyingtotheasewhereÆ=1,wegetforthemodelofequation(1.3)thatlog2hasaninvariantN(b1b2;12b2)distribution.Forthemodelofequation(1.4)wendthat2hasaninvariant(2b1;2b2)distri-butionandforthemodelofequation(1.5)theinvariantdistributionof2isinversegammawithdensitywhihisproportionaltoy7!e2b1=yy2b22.Theseexamplesshowthatsomeofthemodelsthatareusedtodesribethevolatilitydisplayratherdierentinvariantdistributions.Thisobserva-tionsupportsourpointofviewthatnonparametriproeduresarebyallmeanssensibletoolstogetsomeinsightinthebehaviourofthevolatility.Moreover,alltheinvariantdistributionsgivenintheexamplesaboveareunimodal.Sineitisknownthatvolatilitylusteringisanoftenour-ringphenomenon,itishardtobelievethatthisanbeexplainedbyanyofthesemodels.Instead,onewouldexpetinsuhaaseforinstaneabimodaldistributionwithpeaksatertainlevelsoflowandhighvolatility.Nonparametridensityestimationouldperhapsrevealsuhashapeoftheinvariantdensityofthevolatility.Ourmaingoalisestimatingthedensityfifwedon’thaveaontinuousreordofobservationsofS,butweonlyobserveSatthedisretetimeinstants,sayattimes0,,2;:::;n.Toillustratetheunderlyingideas,weonsider(1.1)butrstwithoutthedriftterm,soweassumetohavedSt=tdWt;S0=0:Let’skeepthetimebetweentheobservationsxed(butsmall)forthemoment.Fori=1;2;:::wework,likeinGenon-Catalotetal.(1998,1999),withthenormalizedinrementsXi=1p(SiS(i1)):Forsmall,wehavetheroughapproximationXi=1pZi(i1)tdWt(i1)1p(WiW(i1))=(i1)Zi;3wherefori=1;2;:::wedeneZi=1p(WiW(i1)):BytheindependeneandstationarityofBrownianinrements,thesequeneZ1;Z2;:::isani.i.d.sequeneofstandardnormalrandomvariables.More-over,thesequeneisindependentoftheproessbyassumption.Itisthereforeusefultorstanalyzedisretetimemodelsthatexhibitasimilarstruture