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1、FinancialEconometrics24ParametricEstimationofJump-Diffusions*Keywords:maximumlikelihoodestimation,jump-diffusions,MonteCarlosimulation,termstructureofinterestratesWeconsidertheproblemofestimatingthecoef?cientsofdiffusionprocesseswithjumps.Theseprocesseshavebeenwidelyemployedfordescribingsudden
2、changesbothinstockpriceandininterestratedynamicsandreproducinghigh-ordermo-mentsofsampledata.Inmostcases,thelackofanalyticalexpressionsfortransitiondensitiesmakestheuseofmaximumlikelihoodmethodsaverychallengingtask.Thiscasedescribesandteststhesimulatedmaximumlikelihood(SML)methodintroducedbyPe
3、dersen(1995)andBrandtandSantaClara(2002).Thistechniqueallowsfortheestimationofawidevarietyofdiffusionprocesses,includingthosewhichlackclosed-formexpressionsforthetransitiondensity.WeillustratetheSMLmethodbydevelopingalgorithmsfortheestimationofbothcontinuousandmixed-jump-diffusionprocesses.The
4、latterprocesshasbeenemployedininterestratemod-ellingbyPiazzesi(2001).ThemainideaunderlyingSMListonumericallyevaluatethetransitionproba-bilitiesoftheprocesscorrespondingtoallpairsofvaluestakenbythestatevariableatconsecutivetimes.Ifadiscretizationofthetime–spaceaxesisproperlyre?ned,theresultingt
5、ransitiondensityapproachesaGaussiandistribution.Thelikelihoodestimatorbecomesareliableapproximationoftheexactmaximumlikelihoodesti-mator,namelytheonestemmingfromtheexact,yetunknown,transitiondensityoftheprocess.Sections24.1and24.2introducetheestimationproblemandillustratethegen-eralmethodology
6、,respectively.Section24.3detailsthealgorithm,whileSect.24.4providestwoapplications.First,weconsidertheCox,IngersollandRoss(1985)model(CIR)forshort-terminterestratedynamics.Sincethetransitiondensityfor?withPiergiacomoSabino.52024ParametricEstimationofJump-Diffusionsthismodelisknowninclosed-form
7、,parameterscanbeestimatedusingtheexactmaximumlikelihoodmethod.Consequently,wecanusethismodelasabenchmarkagainstwhichweevaluatetherelativeperformanceofSML.Second,weadaptSMLtojump-diffusionprocessesandimplementtheresultingalgorithmtothege
