On general financial markets with concave transactions costs
A. Rygiel, L. Stettner
TL;DR
The paper analyzes discrete-time financial markets with concave, volume-dependent transaction costs (including fixed components) by introducing liquidation functions $L_t$, solvency sets $G_t$, and related convex-analytic structures. It establishes precise no-arbitrage (NA) conditions and shows how they relate across different market formulations (e.g., $G_t$, $K_t$, $\bar{K}_t$), along with strong arbitrage (SA) concepts and their equivalences under the assumptions $(gL_0)$ and $(L_1)$. The core contribution is the development of asymptotic arbitrage (AA) theory in this setting, proving that arbitrage in proportional-cost markets implies AA in concave-cost markets and providing a framework to transfer AA across market models. The results are extended to markets with non infinitely divisible assets, showing that arbitrage and AA notions persist in discrete-asset settings and establishing conditions under which AA carries over to $G_t^N$-type markets. Altogether, the work furnishes a rigorous foundation for arbitrage analysis in illiquid markets encountered in currency and real estate, where order-size impacts are concave rather than linear.
Abstract
In the paper we study markets with concave transaction costs which depend in a concave way on the volume of transaction. This is typical situation in the case of small investors, which commonly appears in currency and real estate markets. Sufficient conditions for absence of arbitrage are formulated. New notion of asymptotic arbitrage is introduced and used to study the above mentioned markets.