运用MATLAB的好方法

MAXIMUMLIKELIHOODESTIMATIONOFTHECOX-INGERSOLL-ROSSPROCESS:THEMATLAB

IMPLEMENTATION

KamilKlad´ vko1

DepartmentofStatisticsandProbabilityCalculus,UniversityofEconomics,Prague

andDebtManagementDepartment,MinistryofFinanceoftheCzechRepublic

kladivk@vse.czorkamil.kladivko@mfcr.cz

Abstract

Thesquarerootdi usionprocessiswidelyusedformodelinginterestratesbehaviour.Itisanunderlyingprocessofthewell-knownCox-Ingersoll-Rosstermstructuremodel(1985).Weinvestigatemaximumlikelihoodestimationofthesquarerootprocess(CIRprocess)forinterestratetimeseries.TheMATLABimplementationoftheestimationroutineisprovidedandtestedonthePRIBOR3Mtimeseries.

1CIRProcessforInterestRateModeling

Acontinuous-timemodelin nancetypicallyrestononeormorestationarydi usionprocesses{Xt,t≥0},withdynamicsrepresentedbystochasticdi erentialequations:

dXt=µ(Xt)dt+σ(Xt)dWt,

(1)

where{Wt,t≥0}isastandardBrownianmotion.Thefunctionsµ(·)andσ2(·)are,respectively,thedriftandthedi usionfunctionsoftheprocess.

Thefundamentalprocessininterestratemodelingisthesquarerootprocessgivenbythefollowingstochasticdi erentialequation:

drt=α(µ rt)dt+

√tσdWt,

(2)

wherertistheinterestrateandθ≡(α,µ,σ)aremodelparameters.Thedriftfunctionµ(rt,θ)=α(µ rt)islinearandpossessameanrevertingproperty,i.e.interestratertmovesinthedirectionofitsmeanµatspeedα.Thedi usionfunctionσ2(rt,θ)=rtσ2isproportionaltotheinterestratertandensuresthattheprocessstaysonapositivedomain.

Thesquarerootprocess(2)isthebasisfortheCox,Ingersoll,andRossshort-terminterestratemodel[1]andthereforeoftendenotedastheCIRprocessinthe nancialliterature.

1.1CIRprocessdensities

Ifα,µ,σareallpositiveand2αµ≥σ2holds,theCIRprocessiswell-de nedandhasasteadystate(marginal)distribution.Themarginaldensityisgammadistributed.

Formaximumlikelihoodestimationoftheparametervectorθ≡(α,µ,σ)transitionden-sitiesarerequired.TheCIRprocessisoneoffewcases,amongthedi usionprocesses,wherethetransitiondensityhasaclosedformexpression.Wefollowthenotationgivenin[1]onpage391.Givenrtattimetthedensityofrt+ tattimet+ tis

√vq

p(rt+ t|rt;θ, t)=ce u v()Iq(2),

u

1

(3)

ThisworkwassupportedbygrantofIGAVSEnr.IG410046