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1、257StatisticaNeerlandica(2010)Vol.64,nr.3,pp.257–275doi:10.1111/j.1467-9574.2010.00458.xOldandnewapproachestoLIBORmodelingAntonisPapapantoleon*InstituteofMathematics,TUBerlin,Strassedes17.Juni136,10623Berlin,andQuantitativeProductsLaboratory,DeutscheBankAG,Alexanderstr.5,101
2、78Berlin,GermanyInthisarticle,wereviewtheconstructionandpropertiesofsomepopularapproachestomodelingLIBORrates.Wediscussthefollowingframeworks:classicalLIBORmarketmodels,forwardpricemodelsandMarkov-functionalmodels.WeclosewiththerecentlydevelopedafÞneLIBORmodels.KeywordsandPh
3、rases:LIBORrate,LIBORmarketmodel,forwardpricemodel,Markov-functionalmodel,afÞneLIBORmodel.1IntroductionInterestratemarketsarealargeandimportantpartofglobalfinancialmarkets.ThefigurespublishedbytheBankforInternationalSettlements(BIS)showthatinterestratederivativesrepresentmoret
4、han60%oftheover-the-countermarketsovertheyears,intermsofnotionalamount(Table1).Hence,itisimportanttohavemodelsthatcanadequatelydescribethedynamicsandmechanicsofinterestrates.Thereisanotabledifferencebetweeninterestratemarketsandstockorforeignexchange(FX)markets.Inthelatter,t
5、hereisasingleunderlyingtobemodeled,thestockpriceortheFXrate;whereasininterestratemarketsthereisawholefamilyofunderlyingstobemodeled,indexedbythetimeofmaturity.Thisposesuniquechallengesforresearchersinmathematicalfinanceandhasledtosomefascinatingdevelopments.Theinitialapproach
6、estointerestratemodelingconsideredshortratesorinstanta-neousforwardratesasmodelingobjects,andthendeducedfromthemtradablerates.Morerecently,effectiverates,thatis,tradablemarketrates,suchastheLIBORorswaprate,weremodeleddirectly.Modelsforeffectiveratesconsideronlyadiscretesetof
7、maturitydates,theso-calledtenorstructure,whichconsistsofthedateswhentheseratesarefixed.Areviewofthedifferentapproachestomodelinginterestratesisbeyondthescopeofthisarticle.Therearemanyexcellentbooksavailable,focusingonthetheoreticalandpracticalaspectsofinterestratetheory.Weref
8、erthereader,forexample,toBjörk(2004),MusielaandRutkowski(1997),Filipovic´(2
