Likelihood-based inference for correlated diffusio

arXiv:0711.1595v1[stat.ME]10Nov2007LikelihoodbasedinferenceforcorrelateddiffusionsKonstantinosKalogeropoulos∗UniversityofCambridge,DepartmentofEngineering-SignalProcessingLaboratoryPetrosDellaportasAthensUniversityofEconomicsandBusiness,DepartmentofStatisticsGarethO.RobertsUniversityofWarwick,DepartmentofStatisticsFebruary5,2008AbstractWeaddresstheproblemoflikelihoodbasedinferenceforcorrelateddiffusionpro-cessesusingMarkovchainMonteCarlo(MCMC)techniques.Suchataskpresentstwointerestingproblems.First,theconstructionoftheMCMCschemeshouldensurethatthecorrelationcoefficientsareupdatedsubjecttothepositivedefiniteconstraintsofthediffusionmatrix.Second,adiffusionmayonlybeobservedatafinitesetofpointsandthemarginallikelihoodfortheparametersbasedontheseobservationsisgenerallynotavailable.WeovercomethefirstissuebyusingtheCholeskyfactorisationonthediffusionmatrix.Todealwiththelikelihoodunavailability,wegeneralisethedataaugmentationframeworkofRobertsandStramer(2001Biometrika88(3):603-621)tod−dimensionalcorrelateddiffusionsincludingmultivariatestochasticvolatilitymodels.Ourmethod-ologyisillustratedthroughsimulationbasedexperimentsandwithdailyEUR/USD,GBP/USDratestogetherwiththeirimpliedvolatilities.Keywords:MarkovchainMonteCarlo,Multivariatestochasticvolatility,MultivariateCIRmodel,CholeskyFactorisation.∗Address:TrumpingtonStreet,Cambridge,UK,CB21PZ,Tel:[phone],E-mail:[e-mail]11IntroductionDiffusionprocessesprovideanaturalmodelforphenomenaevolvingcontinuouslyintime.Oneoftheirappealingfeaturesisthattheyaredefinedintermsoftheinstantaneousmeanandvarianceoftheprocess.Specifically,adiffusionxtobeysthedynamicsofthefollowingstochasticdifferentialequation(SDE)dxt=μ(t,xt,θ)dt+σ(t,xt,θ)dwt,(1)drivenbystandardBrownianmotionwt.Thefunctionsμ(.)andσ(.)aretermedasthedriftandthevolatilityofthediffusionrespectively.Throughoutthispaperwesuppressthedependenceonttosimplifythenotation,butthemethodologyisalsoapplicabletotimeinhomogeneousdiffusions.Thediffusionprocessxtiswelldefinedif(1)hasauniqueweaksolution,whichtranslatesintosomeregularityconditions(locallyLipschitzwithalineargrowthbound)onμ(.)andσ(.);seechapter5ofRogersandWilliams(1994)formoredetails.Weaddresstheproblemofmodellingseveraldiffusions,denotedbyx{i}t,i=1,...,d.Eachdiffusionx{i}tmayhaveadriftμ{i}(.)andvolatilityσ{i}(.)ofgeneral,yetknown,form.Wealsoallowforcorrelations,corr(dx{i}t,dx{j}t)=ρij=ρji,i6=j,ontheinstantaneousincrements.Theuseofcross-correlationsisquitecommonwhenmodellingmultivariatetimeseries,astheymaycaptureeffectscausedbycommonfactorsoftheunderlyingstochasticprocesses.Inthispaperweillustrateourmethodologythroughtwoexamplesofcorrelateddiffusions.Thefirstexampletargetsinterestratesandbondpricing.Suchtimeseriesoftenexhibitstronginter-dependencies;forinstance,interestratesmaycorrespondtosimilarbondsbutwithdifferentexpirydates,thusgivingrisetocorrelationsamongthem.InSection5weexamineamultivariateversionoftheCoxetal.(1985)model(CIR),oftenusedforsuchdata.Thesecondexampleconsiderscurrencypairswhichareknowntobecorrelated,possiblyduetothecommoncurrenciestheymayrepresent.Section6containsananalysisonEUR/USDandGBP/USDdata,basedonmultivariateversionsofstochasticvolatilitydiffusions,suchasthemodelofHeston(1993).Inbothexamples,theinclusionofcorrelationsinthemodelisessentialfortworeasons.First,theymayaffecttheparameterestimatesoftheindividualdiffusions,aswellastheirprecision.Second,theyreflectcharacteristicsofthemarketwhichmaybeusefulinthebond/optionpricingprocedure.Weproceedbycombiningthediffusionsx{i}ttogetherintoXt=(x{1}t,...,x{d}t)′(with′2denotingtransposition),sothatXtisad−dimensionalvectorforeachtimet.ThediffusionmatrixofXt,A,denotesitsinstantaneouscovarianceandtakesthefollowingform:A:=σ{1}(.)2ρ12σ{1}(.)σ{2}(.)...ρ1dσ{1}(.)σ{d}(.)ρ12σ{1}(.)σ{2}(.)σ{2}(.)2...ρ2dσ{2}(.)σ{d}(.)............ρd1σ{1}(.)σ{d}(.)ρd2σ{2}(.)σ{d}(.)...σ{d}(.)2(2)ThediffusionprocessXtisdefinedthroughthefollowingmulti-dimensionalSDEdXt=M(Xt,θ)dt+Σ(Xt,θ)dWt,(3)whereWtisad−dimensionalBrownianmotionwithindependentcomponents,withvectorvalueddriftM:[0,+∞)×SX×Θ→ℜdwith[M(.)]i=μ{i}(.),andmatrixvaluedvolatility(alsotermedasdispersionmatrix)Σ(·):[0,+∞)×SX×Θ→ℜd×d,whereSXandΘdenotesthedomainofthediffusionXtandtheparametervectorθrespectively.ThedispersionmatrixΣisasquarerootoftheinstantaneouscovariancematrixA=ΣΣ′.ToensureauniqueweaksolutionforXt,werequireauniqueweaksolutionforeachx{i}tandthematrixAtobepositivedefiniteforallt,Xt,θ.Eachdiffusionx{i}tmaybeobserved,withorwithouterror,atafinitesetofpoints,ormaybeentirelyunobserved.Thediffusionwillbetermedasdirectlyobservedincaseswithexactobservationsonallx{i}t,andpartiallyobservedotherwise.Foreaseofexposition,themethodologyofthispaperisinitiallypresentedfordirectlyobserveddiffusions,andadaptationstopartialobservationregimes,asinmultivariatestochasticvolatilitymodels,areprovidedwhennecessary.Similarly,weconsiderobservationsoftheentirevectorofXtateachtime,althoughthisassumptioncaneasilyberelaxed.Wedenotethetimesofobservationsbytk,k=1,...,n,andthedatawithY=nYk=Xtk=(x{1}tk,...,x{d}tk)′,k=1,