Arbitrage and Equilibrium in Asset Exchange Economies A Surv(2)

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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