Arbitrage and Equilibrium in Asset Exchange Economies A Surv(12)

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

De nition3.5Theeconomyεsatis estheNAPSconditionif

m

i=1Si=φ.

Remark3.1Notethatiftheeconomyεsatis esWerner’snonsatiationcondition,i.e.,Ri\Li=φ, i,thentheNAPSconditiongiveninDe nition3.5abovereducestowerner’soriginalconditiongiveninDe nition3.3.

Lemma3.1Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsaretrue:

1.Foranyi,suchthatRi\Li=φ,wehave:

⊥Si={p∈L⊥i|p·y>0, y∈(Ri∩Li)\{0}}.

002. i=1,···,m,Si= ri(Ri)where(Ri)isthepolarconeofRi.

Pageetal.[48]showthatunder[A.1]-[A.2],[A’.3]andWNS],WNMAholdsifm andonlyifSiW=φ(i.e.,Hart’sconditionholdsifandonlyifWerner’sconditionholds).Allouch[5]extendthisresultbyproving,under[A.1]-[A.2]only,thatWNMAm holdsifandonlyifSi=φ.

i=1i=1

Theorem3.3Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsareequivalent:

1.εsatis esWNMA.

2.εsatis esNAPS.

PageandWooders[45]statethatifLi={0}, i,thenNUBAholdsifandonlym ifSiW=φ.Infact,thisresultisaconsequenceofasharperresult:

i=1

Corollary3.1Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsareequivalent:

1.εsatis esNUBA.

m 2.Si=φ.andthelinearityspacesarelinearlyindependent.

i=1

Remark3.2BytheCorollary3.1thereisanabsenceofarbitrageopportunitiesifandonlyifthereexistsapricesystemlimitingarbitrageopportunitiescontainedintheL⊥ispacesandtherearenoarbitrageopportunitiesinthelinearityspaces.Thus,whenthelinearityspacesareequaltozero,nonemptinessofthesetofno-arbitragem prices(i.e.,Si=φ.)isnecessaryandsu cienttoruleoutarbitrageopportunitiesintheeconomy.

186i=1

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