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1、MAXIMUMLIKELIHOODESTIMATIONOFTHECOX-INGERSOLL-ROSSPROCESS:THEMATLABIMPLEMENTATIONKamilKlad´ıvko1DepartmentofStatisticsandProbabilityCalculus,UniversityofEconomics,PragueandDebtManagementDepartment,MinistryofFinanceoftheCzechRepublickladivk@vse.czorkamil.kladivko@mfcr.czAbstrac
2、tThesquarerootdiffusionprocessiswidelyusedformodelinginterestratesbehaviour.Itisanunderlyingprocessofthewell-knownCox-Ingersoll-Rosstermstructuremodel(1985).Weinvestigatemaximumlikelihoodestimationofthesquarerootprocess(CIRprocess)forinterestratetimeseries.TheMATLABimplementati
3、onoftheestimationroutineisprovidedandtestedonthePRIBOR3Mtimeseries.1CIRProcessforInterestRateModelingAcontinuous-timemodelinfinancetypicallyrestononeormorestationarydiffusionprocesses{Xt,t≥0},withdynamicsrepresentedbystochasticdifferentialequations:dXt=µ(Xt)dt+σ(Xt)dWt,(1)where{W
4、t,t≥0}isastandardBrownianmotion.Thefunctionsµ(·)andσ2(·)are,respectively,thedriftandthediffusionfunctionsoftheprocess.Thefundamentalprocessininterestratemodelingisthesquarerootprocessgivenbythefollowingstochasticdifferentialequation:√drt=α(µ−rt)dt+rtσdWt,(2)wherertistheinterestr
5、ateandθ≡(α,µ,σ)aremodelparameters.Thedriftfunctionµ(rt,θ)=α(µ−rt)islinearandpossessameanrevertingproperty,i.e.interestratertmovesinthedirectionofitsmeanµatspeedα.Thediffusionfunctionσ2(rt,θ)=rtσ2isproportionaltotheinterestratertandensuresthattheprocessstaysonapositivedomain.The
6、squarerootprocess(2)isthebasisfortheCox,Ingersoll,andRossshort-terminterestratemodel[1]andthereforeoftendenotedastheCIRprocessinthefinancialliterature.1.1CIRprocessdensitiesIfα,µ,σareallpositiveand2αµ≥σ2holds,theCIRprocessiswell-definedandhasasteadystate(marginal)distribution.Th
7、emarginaldensityisgammadistributed.Formaximumlikelihoodestimationoftheparametervectorθ≡(α,µ,σ)transitionden-sitiesarerequired.TheCIRprocessisoneoffewcases,amongthediffusionprocesses,wherethetransitiondensityhasaclosedformexpression.Wefollowthenotationgivenin[1]onpage391.Givenrt
8、attimetthedensityofrt+∆tattimet+∆tisvq√−u−vp(rt+∆t
9、rt;θ,∆t)=c
