Large Financial Markets, Discounting, and No Asymptotic Arbitrage

Abstract

For a large financial market (which is a sequence of usual, "small" financial markets), we introduce and study a concept of no asymptotic arbitrage (of the first kind), which is invariant under discounting. We give two dual characterizations of this property in terms of (1) martingale-like properties for each small market plus (2) a contiguity property, along the sequence of small markets, of suitably chosen "generalized martingale measures." Our results extend the work of Rokhlin, Klein, and Schachermayer and Kabanov and Kramkov to a discounting-invariant framework. We also show how a market on [0, infinity) can be viewed as a large financial market and how no asymptotic arbitrage, both classic and in our new sense, then relates to no-arbitrage properties directly on [0, ∞).