3ContingentClaimsMarkets

Ch3.ContingentClaimsMarketsZhenlongZHENGBriefintroductionIntheframeofcompletemarket,welookforwardtoseetheequationp=E(mx)moreintuitive.Thestructureisasfollows:3.1ContingentClaims3.2Risk-NeutralProbabilities3.3InvestorsAgain3.4RiskSharing3.5StateDiagramandPriceFunction3.1ContingentClaimsAcontingentclaimisasecuritythatpaysonedollar(oroneunitoftheconsumptiongood)inonestatesonlytomorrow.pc(s)isthepricetodayofthecontingentclaim.（状态价格）CompletemarketInacompletemarketinvestorscanbuyanycontingentclaim.Theydon’tnecessarilyhavetobefacedwithexplicitcontingentclaims;theyjustneedenoughothersecuritiestospanorsynthesizeallcontingentclaims.（有几种状态，就有几种相互无法复制的证券，就是完全市场）x(s)denoteanasset’spayoffinstateofnaturesSincetheassetisabundleofcontingentclaims,itspricemustthenequalthetotalvalueofthecontingentclaims.p(x)=∑pc(s)x(s)(3.1)Itiseasiertotakeexpectationsratherthansumoverstates.p(x)=∑(s)(pc(s)/(s))x(s),where(s)istheprobabilitythatstatesoccurs.Thendefinemastheratioofcontingentclaimpricetoprobability，m(s)=pc(s)/(s)ConclusionaboutdiscountfactorNowwecanwritethebundlingequationasanexpectation,p=∑(s)m(s)x(s)=E(mx)Iftherearecompletecontingentclaims,adiscountfactorexists,anditisequaltothecontingentclaimpricedividedbyprobabilities.ExpandtoinfinitespaceIngeneral,wepositstatesofnatureωthatcantakecontinuous(uncountablyinfinite)valuesinaspaceΩ.Inthiscase,thesumsbecomeintegrals,andwehavetousesomemeasuretointegrateoverΩ.Thus,scalingcontingentclaimspricesbysomeprobability-likeobjectisunavoidable.3.2Risk-neutralprobabilitiesDefineThe*(s)arepositive,lessthanorequaltooneandsumtoone,sotheyarealegitimatesetofprobabilities.()()()()()()()(),()()()()()()ssfpcsspcspcsmsspcsEmmsssRmssEmThenwecanrewritetheassetpricingformulaas:WeusethenotationE*toremindusthattheexpectationusestheriskneutralprobabilitiesπ*insteadoftherealprobabilitiesπ.()()()()()()1()()()ssffpxpcsxsmssxsExsxsRR——风险中性定价*(s)=(m(s)/E(m))(s)*givesgreaterweighttostateswithhigherthanaveragemarginalutilitymriskaversionisequivalenttopayingmoreattentiontounpleasantstates,relativetotheiractualprobabilityofoccurrence.Applicationtoreportone’sreasonablesubjectiveprobabilities.Wecanalsothinkofthediscountfactormasthederivativeorchangeofmeasurefromtherealprobabilitiesπtothesubjectiveprobabilities*连续时间在完全市场中，两者的风险源相同。From(1.35),wehave()()fpttdpDddpEdtrdtEdtppp——超额收益可见是风险价格。1.34ppfdtdzdrdtdzdpp（）风险中性定价在风险中性世界，风险价格必须等于0，即0),ffpfpppdrdtdpDrdtdzppDrp＝。这样，在风险中性世界中，＝（＋由于＝－也就是说，我们只要把价格的偏移率减少，并去掉随机贴现因子的扰动项就可以得到风险中性世界的随机过程。3.3Investors’choiceTheinvestorstartswithapileofinitialwealthyandastate-contingentincomey(s).Hepurchasescontingentclaimstoeachpossiblestateinthesecondperiod.Hisproblemisthen,()()()[].()()()()ccssssMaxucsucsstcpcscsypcsysEliminatingtheLagrangemultiplierλ,·Coupledwithp=E(mx),weobtaintheconsumption-basedmodelagain.(())()()()()(())()()()ucspcssucpcsucsmssucmarginalratesofsubstitutionTheinvestor’sfirstorderconditionssaythatmarginalratesofsubstitutionbetweenstatestomorrowequalstherelevantpriceratio,边际替代率相对价格比（经概率调整）1122()(())()(())msucsmsucsEconomicsbehindthisapproachtoassetpricing(figure3.1)3.4RiskSharingIncompletemarkets,thepricesarethesameforallinvestors.如果信息是透明的，每个人都知道客观概率，则marginalutilitygrowthshouldbethesameforallinvestorsIfinvestorshavethesamehomotheticutilityfunction(forexample,powerutility),thenconsumptionitselfshouldmoveinlockstep.11()()()()ijijttijttucucucuc11ijttijttccccItmeansthatshockstoconsumptionareperfectlycorrelatedacrossindividuals.Itdoesn’tsaythatexpectedconsumptiongrowthshouldbeequal;itsaysthatconsumptiongrowthshouldbeequalexpost.Inacompletecontingentclaimsmarket,allinvestorsshareallrisks,sowhenanyshockhits,ithitsusallequally(afterinsurancepayments).Pareto-optimalrisksharing.Supposeasocialplannerwishedtomaximizeeveryone’sutilitygiventheavailableresources.Forexample,withtwoinvestorsiandj,hewouldmaximizemax()()..titjitjtttijatttEucEucstcccfirstorderconditionThissimplefacthasprofoundimplications:Itshowsyouwhyonlyaggregateshocksshouldmatterforriskprices.Anyidiosyncraticincomeriskwillbeequallyshared,andso1/Nofitbecomesanaggregateshock.Thenthestochasticdiscountfactorsmthatdetermineassetpricesarenolongeraffectedbytrulyidiosyncraticrisks.Muchofthissensethatonlyaggregateshocksmatterstayswithusinincompletemarketsaswell.()()iijjttucucSub-marketsforrisksharing:InsurancemarketbondmarketstockmarketReasonsforindividualconsumptionsnotmoveinlockstep:(1)Therealeconomydoesnotyethavecompletemarketsorfullrisksharing.(2)Differentutilityfunctions.(3)Differentvaluesofindividualimpatientcoefficients.(4)Different3.5StateDiagramandPriceFunctionThinkofthecontingentclaimspricepcandassetpayoffsxasvectorsinRS,whereeachelementgivesthepriceorpayofftothecorrespondingstate,pc=[pc(1)pc(2)…pc(S)]’x=[x(1)x(2)…x(S)]’Figure7isagraphofthesevectorsinRS图7状态价格与回报P=0(超额收益率RfP=2P=1（收益率）状态1回报状态2回报1/Rfpc线的斜率在经概率调整后的状态偏好中性世界中，pc(1)=pc(2),因此pc线是45度线。在现实生活中，投资者