In discrete time markets with proportional transaction costs, Schachermayer [Sch04] shows 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, Rásonyi, and Stricker [KRS02]. The prospective strict no-arbitrage condition is slightly weaker than robust no-arbitrage, 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 [Sch04], the consistent frictionless prices may lie on the boundary of the bid-ask spread.On the technical level, a crucial difference to Schachermayer [Sch04] and Kabanov-Rásonyi-Stricker [KRS03] is that we prove closedness without having at hand that the nullstrategies form a linear space.

A standing assumption in the literature on proportional transaction costs is efficient friction. Together with robust no free lunch with vanishing risk, it rules out strategies of infinite variation as they usually appear in frictionless markets. In this paper, we show how the models with and without transaction costs can be unified.The bid and ask prices of a risky asset are given by càdlàg processes which are locally bounded from below and may coincide at some points. In a first step, we show that if the bid–ask model satisfies “no unbounded profit with bounded risk” for simple strategies, then there exists a semimartingale lying between the bid and ask price processes.In a second step, under the additional assumption that the zeros of the bid–ask spread are either starting points of an excursion away from zero or inner points from the right, we show that for every bounded predictable strategy specifying the amount of risky assets, the semimartingale can be used to construct the corresponding self-financing risk-free position in a consistent way. Finally, the set of most general strategies is introduced, which also provides a new view on the frictionless case.