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

http://www.77cn.com.cningthebasicgeometryof

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@http://www.77cn.com.cn

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