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

In discrete-time markets with proportional transaction costs, Schachermayer (Math. Financ. 14:19-48, 2004) showed that robust no-arbitrage is equivalent to the existence of a strictly consistent price system. In this paper, we introduce the concept of prospective strict no-arbitrage that is a variant of the strict no-arbitrage property from Kabanov et al. (Finance Stoch. 6:371-382, 2002). The prospective strict no-arbitrage condition is slightly weaker than the robust no-arbitrage condition, and it implies that the set of portfolios attainable from zero initial endowment is closed in probability. A weak version of prospective strict no-arbitrage turns out to be equivalent to the existence of a consistent price system. In contrast to the fundamental theorem of asset pricing of Schachermayer (Math. Financ. 14:19-48, 2004), the consistent frictionless prices may lie on the boundary of the bid-ask spread. On the technical level, a crucial difference to Schachermayer (Math. Financ. 14:19-48, 2004) and Kabanov et al. (Finance Stoch. 7:403-411, 2003) is that we prove closedness without having at hand that the null-strategies form a linear space.