Large Financial Markets and Asymptotic Arbitrage with Small Transaction Costs

2012/11/02 by Klein, Irene, Lepinette, Emmanuel, Ostafe, Lavinia
#60G44 #91B24 #91B70 #FOS: Economics and business #FOS: Mathematics #Pricing of Securities (q-fin.PR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1211.0443

Abstract

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for large financial markets with small proportional transaction costs \la_n on market n in terms of contiguity properties of sequences of equivalent probability measures induced by \la_n--consistent price systems. These results are analogous to the frictionless case. Our setting is simple, each market n contains two assets with continuous price processes. The proofs use quantitative versions of the Halmos--Savage Theorem and a monotone convergence result of nonnegative local martingales. Moreover, we present an example admitting a strong asymptotic arbitrage without transaction costs; but with transaction costs \la_n>0 on market n (\la_n→0 not too fast) there does not exist any form of asymptotic arbitrage.

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