EXPONENTIAL-POLYNOMIAL FAMILIES AND THE TERM STRUC

EXPONENTIAL-POLYNOMIALFAMILIESANDTHETERMSTRUCTUREOFINTERESTRATESDAMIRFILIPOVI´CDEPARTMENTOFMATHEMATICS,ETH,R¨AMISTRASSE101,CH-8092Z¨URICH,SWITZERLAND.E-MAIL:[e-mail]-polynomialfamiliesliketheNelson–SiegelorSvens-sonfamilyarewidelyusedtoestimatethecurrentforwardratecurve.Weinvestigatewhetherthesemethodsgowellwithinter-temporalmodelling.WecharacterizetheconsistentItˆoprocesseswhichhavethepropertytoprovideanarbitragefreeinterestratemodelwhenrepresentingtheparametersofsomeboundedexponential-polynomialtypefunction.Thisincludesinparticulardiffusionprocesses.Weshowthatthereisastronglimitationontheirchoice.Boundedexponential-polynomialfamiliesshouldrathernotbeusedformod-ellingthetermstructureofinterestrates.Keywords:consistentItˆoprocess,diffusionprocess,exponential-polynomialfamily,forwardratecurve,interestratemodel,inverseproblem1.IntroductionThecurrenttermstructureofinterestratescontainsallthenecessaryinformationforpricingbonds,swapsandforwardrateagreementsofallmaturities.Itisusedfurthermorebythecentralbanksasindicatorfortheirmonetarypolicy.Thereareseveralalgorithmsforconstructingthecurrentforwardratecurvefromthe(finitelymany)pricesofbondsandswapsobservedinthemarket.Widelyusedaresplinesandparameterizedfamiliesofsmoothcurves{F(.,z)}z∈Z,whereZ⊂RN,N≥1,denotessomefinitedimensionalparameterset.ByanoptimalchoiceoftheparameterzinZanoptimalfitoftheforwardcurvex7→F(x,z)totheobserveddataisattained.Herex≥0denotestimetomaturity.InthatsensezrepresentsthecurrentstateoftheeconomytakingvaluesinthestatespaceZ.ExamplesaretheNelson–Siegel[8]familywithcurveshapeFNS(x,z)=z1+(z2+z3x)e−z4xandtheSvensson[11]family,anextensionofNelson–Siegel,FS(x,z)=z1+(z2+z3x)e−z5x+z4xe−z6x.Table1givesanoverviewofthefittingproceduresusedbysomeselectedcentralbanks.ItistakenfromthedocumentationoftheBankforInternationalSettle-ments[1].DespitetheflexibilityandlownumberofparametersofFNSandFS,theirchoiceissomewhatarbitrary.Weshalldiscussthemfromaninter-temporalpointofview:Alotofcross-sectionaldata,i.e.dailyestimationsofz,isavailable.ThereforeitDate:September11,1998(firstdraft);December9,1999(thisdraft).12DAMIRFILIPOVI´CTable1.ForwardratecurvefittingprocedurescentralbankcurvefittingprocedureBelgiumNelson–Siegel,SvenssonCanadaSvenssonFinlandNelson–SiegelFranceNelson–Siegel,SvenssonGermanySvenssonItalyNelson–SiegelJapansmoothingsplinesNorwaySvenssonSpainNelson–Siegel(before1995),SvenssonSwedenSvenssonUKSvenssonUSAsmoothingsplineswouldbenaturaltoaskforthestochasticevolutionoftheparameterzovertime.Butthenthereexisteconomicconstraintsbasedonnoarbitrageconsiderations.Following[2],insteadofFNSandFSweconsidergeneralexponential-polynomialfamiliescontainingcurvesoftheformF(x,z)=KXi=1niXμ=0zi,μxμe−zi,ni+1x.Hencelinearcombinationsofexponentialfunctionsexp(−zi,ni+1x)oversomepoly-nomialsofdegreeni∈N0.ObviouslyFNSandFSareofthistype.WereplacethenzbyanItˆoprocessZ=(Zt)t≥0takingvaluesinZ.Thefollowingquestionsarise:•DoesF(.,Z)provideanarbitragefreeinterestratemodel?•AndwhataretheconditionsonZforit?WorkingintheHeath–Jarrow–Morton[5]–henceforthHJM–frameworkwithdeterministicvolatilitystructure,Bj¨orkandChristensen[2]showedthattheexpo-nential-polynomialfamiliesareinacertainsensetoolargetocarryaninterestratemodel.ThisresulthasbeengeneralizedfortheNelson–Siegelfamilyin[4],includingstochasticvolatilitystructure.Expandingthemethodsusedinthere,wegiveinthispaperthegeneralresultforboundedexponential-polynomialfamilies.Thepaperisorganizedasfollows.InSection2weintroducetheclassofItˆoprocessesconsistentwithagivenparameterizedfamilyofforwardratecurves.Con-sistentItˆoprocessesprovideanarbitragefreeinterestratemodelwhendrivingtheparameterizedfamily.TheyarecharacterizedintermsoftheirdriftanddiffusioncoefficientsbytheHJMdriftcondition.BysolvinganinverseproblemwegetthemainresultforconsistentItˆoprocesses,statedinSection3.Itisshownthattheyareremarkablylimited.Theproofisdividedintoseveralsteps,giveninSections4,5and6.InSection7weextendthenotionofconsistencytoe-consistencywhenPisnotamartingalemeasure.Themainresultreadsmuchclearerwhenrestrictedtodiffusionprocesses,asshowninSection8.Itturnsoutthate-consistentdiffusionprocessesdrivingEXPONENTIAL-POLYNOMIALFAMILIES3boundedexponential-polynomialfamilieslikeNelson–SiegelorSvenssonareverylimited:mostofthefactorsareeitherconstantordeterministic.ItisshowninSection9,thatthereisnonon-trivialdiffusionprocesswhichise-consistentwiththeNelson–Siegelfamily.Furthermoreweidentifythediffusionprocesswhichise-consistentwiththeSvenssonfamily.Itcontainsjustonenondeterministiccom-ponent.ThecorrespondingshortratemodelisshowntobethegeneralizedVasicekmodel.Weconcludethatboundedexponential-polynomialfamilies,inparticularFNSandFS,shouldrathernotbeusedformodellingthetermstructureofinterestrates.2.ConsistentItˆoprocessesForthestochasticbackgroundandnotationswereferthereaderto[9]and[6].Let(Ω,F,(Ft)0≤t∞,P)beafilteredcompleteprobabilityspace,satisfyingtheusualconditions,andletW=(W1t,...,Wdt)0≤t∞denoteastandardd-dimensional(Ft)-Brownianmotion,d≥1.LetZ=(Z1,...,ZN)denote