Arbitrage and Equilibrium in Asset Exchange Economies A Surv

Abstract: This article surveys some recent progress on arbitrage and equilibrium in asset exchange economies. Using the basic geometry of arbitrage, the relationships between various no-arbitrage conditions appeared in the literature are presented. The rel

AMO-AdvancedModellingandOptimization,Volume5,Number3,2003

ArbitrageandEquilibriuminAssetExchange

Economies:ASurvey1

HongbinDong,GuangyaChenandShouyangWang2

InstituteofSystemsScience,AcademyofMathematicsandSystemsSciences,

ChineseAcademyofSciences,Beijing100080,China

Abstract:Thisarticlesurveyssomerecentprogressonarbitrageand

ingthebasicgeometryof

arbitrage,therelationshipsbetweenvariousno-arbitrageconditionsap-

pearedintheliteraturearepresented.Therelationshipsbetweensome

ofthebasicno-arbitrageconditionstogetherwiththeexistenceresultof

anequilibriuminDanaetal.[16]provideanoverviewofsu cientcon-

ditionsforequilibrium.Undercertainconditionsthevariousno-arbitrage

conditionsintheliteratureareequivalentandnecessaryandsu cient

fortheexistenceofanequilibrium.

Keywords:Arbitrage;Equilibrium;Compactness;Cones;Uniformity

1Introduction

SincethepioneeringcontributionsofGrandmont[21,22,23],Green[25],andHart

[27],therelationshipbetweenarbitrageandequilibriuminassetexchangeeconomiesallowingshortsaleshasbeenoneofthehotsubjectineconomicand nancialstud-ies.Anarbitrageopportunityisamutuallycompatiblesetofnettradeswhichareutilitynondecreasingand,atmost,costlesstomake.Conditionsthatlimitutil-ityarbitragearecentraltoestablishingexistenceingeneralequilibriummodelsofexchangeeconomieswithunboundedshortsales(see,forexample,Page[40]andWerner[55]).Whenunboundedshortsalesareallowed,asisnaturalinassetmar-ketmodels,agents’choicersetsareboundedfrombelow,andasaconsequence,unboundedandmutuallycompatiblearbitrageopportunitiescanbeexhaustedmayfailtoexist,andthus,equilibriummayfailtoexist.Byassumingthatmarketsadmit“noarbitrage”,theeconomycanbeboundedendogenously butthisisnot1

2SupportedbyNSFC,CASandMADIS.Thecorrespondingauthor,Email:sywang@

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Abstract: This article surveys some recent progress on arbitrage and equilibrium in asset exchange economies. Using the basic geometry of arbitrage, the relationships between various no-arbitrage conditions appeared in the literature are presented. The rel

enoughforexistence.SincetheseminalcontributionsofWerner[55],muchoftheresearchonassetmarketmodelshavefocuseduponconditionslimitingarbitrage(i.e.,no-arbitrageconditions)andupontherelationshipbetweensuchconditionsandtheexistenceofanequilibrium.

No-arbitrageconditionsappearedinliteraturegenerallyfallintothreebroadcategories:

(i)Conditionsonnettrades,forexample,Hart[27],Page[41],Nielsen[37],Allouch[2],Pageetal.[48]andAllouch[4].

(ii)Conditionsonprices,forexample,Green[25],Grandmont[23,24],Ham-mond[26]andWerner[55].

(iii)Conditionsonthesetofutilitypossibilities(namely,compactness),forexample,BrownandWerner[8]andDanaetal.[16].

Inatemporaryequilibriummodel,Grandmont[23]showsthattheoverlappingexpectationsconditionsisnecessaryandsu cientfortheexistenceofanequilib-rium.Grandmont’sresultisthe rsttogivenecessaryandsu cientconditionsforexistenceofequilibriuminaneconomicsmodelwithassettradingandunboundedshortsales.Grandmont’sresultcontinuestoholdinanassetmarketsettingwithunrestrictedshortselling,providedeachinvestor’sasymptoticrisktoleranceiszero.Inparticular,Hart[27],Milne[34],Hammond[26]andPage[38,41,42]showthatoverlappingexpectationsissu cientfortheexistenceofanequilibriuminanassetmarketmodelinwhicheachinvestor’sasymptoticrisktoleranceiszero.Theyalsoshowthattheequilibriumpricevectormustbecontainedintheoverlapofinvestorexpectations(seePage[39]andHammond[26]).Thus,itfollowsasacorollaryoftheresultsofHart[27],Hammond[26],andpage[39])thatifeachinvestor’spref-erencearenotdependentonpricesandeachinvestor’sasymptoticrisktoleranceiszero,thenoverlappingexpectationsisnecessaryandsu cientfortheexistenceofanequilibrium(seealsoMilne,[34]).Page[44]generalizestheoverlappingexpecta-tionsconditionandshowsthisgeneralizedconditionisnecessaryandsu cientfortheexistenceofanequilibriuminanassetmarketmodelinwhichpreferencesarenotdependentonpricesandinvestors’sareallowedtohaveasymptoticrisktolerancesgreaterthanzero.

Attheoppositeendofthespectrumfromthemodelsoftemporaryequilibriumandincompletemarketsarethegeneralequilibriummodelsofexchangeeconomieswithunboundedshortsales(see,forexample,page[40];Werner[55];Niesen[37];PageandWooders[43,45]andChichilnisky[11]).Theroleplayedbyconditionslimitingarbitrageingeneralequilibriummodelswithshortsalesistoboundtheeconomyendogenously.

Forexample,Hart[27]introducestheweakno-market-arbitrageconditiononnettradeswhichrequiresthatallmutuallycompatiblearbitrageopportunitiesbeuseless.Hart’s[27]conditionofweak-no-market-arbitrageholdsifandonlyiftheprojectionofsetofrationalallocationsupontheCartesianproductoftheagents’

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Abstract: This article surveys some recent progress on arbitrage and equilibrium in asset exchange economies. Using the basic geometry of arbitrage, the relationships between various no-arbitrage conditions appeared in the literature are presented. The rel

subspacesofusefulnettradesiscompact.Ifinaddition,weaklyuniformissatis ed,thenHart’sconditionalsoimpliesthecompactnessofthesetofrationalutilitypossibilities.

Werner[55]introducestheno-arbitragepricesystemconditiononpriceswhichrequiresthattherebeanonemptysetofpricessuchthateachpricecontainedinthisnon-emptysubsetassignsapositivevaluetoanyvectorofusefulnettradesbelongingtoanyagent.Wernerthenassumesthatforeachagentthesetofusefulnettradesatendowmentsisnon-empty.Werner’s[55]conditionofno-arbitragepricesystemimpliesdirectlythecompactnessofthesetofutilitypossibilitiesbutallowthesetofrationalallocationstobeunbounded.AnespeciallyintriguingaspectofWerner’sexistenceresultisthatitdoesnotrequirelo …… 此处隐藏：5909字，全部文档内容请下载后查看。喜欢就下载吧 ……