Convex duality with transaction costs

TL;DR

This paper establishes convex duality results for super-replication problems in continuous-time markets with proportional transaction costs, showing equivalence of model-independent and model-specific approaches under certain conditions.

Contribution

It proves the equivalence of two super-replication problems with transaction costs using convex duality, highlighting the impact of transaction costs on model reliance.

Findings

Super-replication problems have the same value under certain conditions.

Convex duality links model-independent and model-specific super-replication.

Transaction costs prevent reduction of super-replication costs via model structure.

Abstract

Convex duality for two two different super--replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one the problems considered is the model--independent hedging that requires the super--replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super--replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super--replication cost.