Aggregation

AggregationandmodelconstructionforvolatilitymodelsOleE.Barndorff-NielsenDepartmentofMathematicalSciences,UniversityofAarhus,NyMunkegade,DK-8000AarhusC,[e-mail]ffieldCollege,OxfordOX11NF,[e-mail]1998AbstractInthispaperwewillrigourouslystudysomeofthepropertiesofcontinuoustimestochasticvolatilitymodels.Wehavefivemainresults:(i)thestochasticvolatil-ityclasscanbelinkedtoCoxprocessbasedmodelsoftick-by-tickfinancialdata;(ii)wecharacterisethemoments,autocorrelationfunctionandspectrumofsquaredreturns;(iii)basedonlyondiscretetimereturns,wegiveasimpleconsistentandasymptoticallynormallydistributedestimatorofcontinuoustimevolatilitymodelswithoutanysimulationordiscretisationerror.Furthermore,wereviewanewclassofOrnstein-Uhlenbeckprocessesofvolatility,introducedinacompanionpaper,whichallows(iv)thediscretetimereturnstobesimulatedwithoutanyformofdis-cretisationerror,(v)explicitmodellingofcorrelationstructuresandallowanalyticcalculationsofthepropertiesofreturns.1Contents1Introduction32Basicmodeltype42.1Notationandcharacterisation.........................42.2Subordination..................................92.3AggregationalGaussianity...........................102.4Stochasticvolatilityandtick-by-tickdata...................103Returns113.1Variousmoments................................113.2Quasi-likelihoodestimationofdynamics...................183.3Multivariateversions..............................194Ornstein-Uhlenbecktypevolatilities214.1Motivation....................................214.2BackgrounddrivingL´evyprocess.......................224.3Existence....................................234.4L´evydensities..................................234.5IntegralsoftheBDLP.............................244.6GeneralizedinverseGaussiandistributions..................254.6.1Generalcase...............................254.6.2InverseGaussiandistribution.....................264.6.3Positivehyperbolicdistribution....................274.6.4Inversegammadistribution......................274.6.5Gammadistribution..........................274.7Alternativemodellingapproach........................285Conclusion296Appendix:alemmaandsomeproofs307Acknowledgements3221IntroductionContinuoustimestochasticvolatility(SV)modelshavehadasubstantialimpactontheor-eticalfinancialeconomicsandeconometrictheoryandpractice.AreviewoftheliteratureisgiveninGhysels,Harvey,andRenault(1996).Astandardmodelfortheevolutionofanassetpriceintheliteratureiswherex∗(t)isthelog-priceandw(t)isBrownianmotion,thenx∗(t)followsthesolutiontoalinearstochasticdifferentialequation(SDE)oftheformdx∗(t)=μ+βσ2(t) dt+σ(t)dw(t),wheret≥0andwhereσ2(t),theinstantaneousvolatility,islatent.Suchmodels,byappro-priatedesignofthestochasticprocessforσ2(t),allowaggregatedreturns{yn}measuredoveraperiodΔ,whereyn=ZnΔ(n−1)Δdx∗(t)=x∗(nΔ)−x∗{(n−1)Δ},Δ0,tobeheavy-tailed,exhibitvolatilityclusteringandaggregatetoGaussianityasΔgetslarge.Thesearethemainfeaturesofassetreturnssurveyedby,forexample,Campbell,Lo,andMacKinlay(1997,pp.17-21).Commonmodelsforσ2(t)includeanOrnstein-UhlenbeckprocesswithBrownianmotionincrementswrittenforlogσ2(t)(e.g.HullandWhite(1987)),anARCHdiffusion(Nelson(1990))andasquarerootprocess(e.g.Heston(1993)).Inthisparagraphwewillassume{σ2(t)}isindependentofthe{w(t)}.Thenwhateverthemodelforσ2itfollowsthatyn|σ2n∼N(μΔ+βσ2n,σ2n).whereσ2n=σ2∗(nΔ)−σ2∗{(n−1)Δ},andσ2∗(t)=Zt0σ2(u)du.Thisimpliesintegratedvolatilityplaysacrucialroleincontinuoustimevolatilitymodels.Unfortunately,existingmodelsofvolatilitydonotallowaneasytreatmentofintegratedvolatilityandsomostresearcherstendtoresorttodiscretisationapproximationseventosimulatereturnsequences.Inthispaperwewillrigourouslystudysomeofthepropertiesofcontinuoustimestochasticvolatilitymodels.Wehavefivemainresults:(i)stochasticvolatilityclasscanbelinkedtoCoxprocessbasedmodels,putforwardedindependentlybyRogersandZane(1998)andRydbergandShephard(1998),oftick-by-tickfinancialdata;(ii)themoments,autocorrelationfunctionandspectrumof{y2n}arecharacterised;(iii)basedonlyondis-cretetimereturns,wegiveasimpleconsistentandasymptoticallynormallydistributedestimatorofcontinuoustimevolatilitymodelswithoutanysimulationordiscretisation3error.Furthermore,wereviewanewclassofOrnstein-Uhlenbeckprocessesofσ2(t)in-troducedinacompanionpaperBarndorff-NielsenandShephard(1998),whichallows(iv)the{yn}tobesimulatedwithoutanyformofdiscretisationerroraswellasproviding(v)simpleandinterpretabledynamicstructuresforthevolatilityallowingexplicitmodellingofcorrelationstructuresandanalyticcalculationsofthepropertiesofreturns.Thestructureofthepaperisasfollows.InSection2wewillformallyintroduceournotationandconsidersomeofthebasicmathematicalpropertiesofx∗(t).Inthissectionwealsoconnectstochasticvolatilitymodelswiththerecentworkontick-by-tickmodelswherepointprocesseshavebeenusedtomodelthetimebetweentrades(see,forexample,EngleandRussell(1998),Ghysels,Jasiak,andGourieroux(1998),RydbergandShephard(1998)andRogersandZane(1998)).InparticularweshowthatwecanderivetheSVclassofprocessesasali