The super-replication theorem under proportional transaction costs revisited

TL;DR

This paper revisits the super-replication theorem in a market with transaction costs, establishing duality results under different admissibility conditions and clarifying the dual processes involved.

Contribution

It provides two versions of the super-replication theorem linking primal admissibility conditions to dual martingale properties, enhancing theoretical understanding.

Findings

Two versions of the super-replication theorem are established.

Connections between primal admissibility and dual martingale types are clarified.

The results extend the classical theorem to markets with proportional transaction costs.

Abstract

We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under transaction costs and the associated dual problem. We give two versions of this theorem. The first theorem relates a num\'eraire-based admissibility condition in the primal problem to the notion of a local martingale in the dual problem. The second theorem relates a num\'eraire -free admissibility condition in the primal problem to the notion of a uniformly integrable martingale in the dual problem.