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In discrete time markets with proportional transaction costs, Schachermayer(2004) shows that robust no-arbitrage is equivalent to the existence of astrictly consistent price system.  In this paper, we introduce the concept of prospective strict no-arbitragethat is a variant of the strict no-arbitrage property from Kabanov, R\'asonyi,and Stricker (2002). The prospective strict no-arbitrage condition is slightlyweaker than robust no-arbitrage, and it implies that the set of portfoliosattainable from zero initial endowment is closed in probability. A weak versionof prospective strict no-arbitrage turns out to be equivalent to the existenceof a consistent price system. In contrast to the fundamental theorem of assetpricing of Schachermayer (2004), the consistent frictionless prices may lie onthe boundary of the bid-ask spread.  On the technical level, a crucial difference to Schachermayer (2004) andKabanov-R\'asonyi-Stricker (2003) is that we prove closedness without having athand that the null-strategies form a linear space.

Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs