On the quasi-sure superhedging duality with frictions

TL;DR

This paper establishes a duality for superhedging in discrete-time markets with transaction costs and model uncertainty, removing previous restrictive assumptions and extending to constrained portfolios.

Contribution

It proves the superhedging duality under No Strict Arbitrage, relaxing earlier No Arbitrage of the Second Kind assumptions, and extends results to models with portfolio constraints.

Findings

Duality holds under No Strict Arbitrage.

Removes the No Arbitrage of the Second Kind assumption.

Extends to models with portfolio constraints.

Abstract

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchard, B., Deng, S. and Tan, X., Super-replication with proportional transaction cost under model uncertainty, Math. Fin., 29(3):837-860, 2019] and to show the duality under the more natural condition of No Strict Arbitrage. In addition, we extend the results to models with portfolio constraints.