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1、Chapter1ABenchmarkApproachtoRiskManagementToprovideforthisbookarelevantÞeldofapplicationforfunctionalsofmultidimen-sionaldiffusions,problemsofpricingandhedgingwillbediscussedinageneralÞnancialmodelingframework.ThischapterintroducesauniÞedcontinuoustimeframeworkforÞna
2、ncialandinsurancemodeling.Itcanbeappliedtoportfolioopti-mization,derivativepricing,Þnancialmodeling,actuarialpricingandriskmeasure-ment.ItisbasedonthebenchmarkapproachpresentedinPlatenandHeath(2010)andtheconceptofbenchmarkedriskminimization,seeDuandPlaten(2012a).Theb
3、estperforming,strictlypositiveportfolioischosenasnaturalbenchmarkforassetallocationandalsoasnaturalnumraireforpricing.Thisportfolioisthegrowthoptimalportfolio(GOP),whichmaximizesexpectedgrowthorlog-utility.Further-more,itisalsothenumraireportfolio(NP)suchthatanynonne
4、gativeportfolio,denominatedinunitsoftheNPturnsouttobeasupermartingale.Thisfundamentalpropertyleadstoanaturalpricingruleundertherealworldprobabilitymeasure,whichidentiÞestheminimalreplicatingprice.Wealertthereadertoanimportantpropertyofthebenchmarkapproach,namelythata
5、nequivalentriskneutralprob-abilitymeasureneednotexist.ThisprovidesthemodelerwithsigniÞcantfreedomcomparedtotheclassicalriskneutralapproach.Notonlymodelswillbeaccom-modatedthatarecoveredbymostoftheclassicalno-arbitragepricingliteratureinÞnancialmathematics,including,f
6、orinstance,KaratzasandShreve(1998)andBjrk(1998),butalsoamuchwiderrangeofmodelswillbeallowed,whichgobe-yondtheclassicalriskneutralparadigm,seee.g.LoewensteinandWillard(2000),FernholzandKaratzas(2005),Platen(2002),KaratzasandKardaras(2007),andGalessoandRunggaldier(2010
7、).Thefocusofthisbookwillbeontractablemod-els,whichmaygobeyondtheclassicalno-arbitrageworld,andthecomputationoftheirfunctionals.1.1AContinuousMarketModelManyapplicationswewilldiscussinvolveÞnancialinstrumentsinacontinu-ousmarket,wherepricesoftradedsecuritiesdonotexhib
8、itjumpsandtheirJ.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications1toFinance,Bocconi&SpringerSeries5,DOI10.
