Semimartingale price systems in models with transaction costs beyond efficient friction

by Christoph Kühn1 and Alexander Molitor2
1Institut für Mathematik, Goethe-Universität Frankfurt, D-60054 Frankfurt a.M., Germany
2Institut für Mathematik, Goethe-Universität Frankfurt, D-60054 Frankfurt a.M., Germany


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 the ask price 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 the ask price process. 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.

Key words:

Proportional transaction costs, No unbounded profit with bounded risk, Strategies of infinite variation, Semimartingales, Stochastic Integration
JEL Classification:  G12
Mathematics Subject Classification (2020):  91G10, 60H05, 26A42, 60G40