Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality

TL;DR

This paper introduces a new duality framework linking Entropy Martingale Optimal Transport with a nonlinear pricing-hedging problem, extending classical optimal transport by incorporating martingale constraints and less restrictive penalties.

Contribution

It develops a novel duality between entropy martingale optimal transport and a nonlinear optimization problem with financial interpretation, broadening the scope of robust pricing-hedging theories.

Findings

Established a nonlinear robust pricing-hedging duality.

Analyzed the impact of penalty variations on duality.

Demonstrated convergence of EMOT to MOT under certain conditions.

Abstract

The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al.. The Problem (A) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive penalization terms $\mathcal{D}_{U}$, which may not have a divergence formulation. In Problem (B) the objective functional, associated via Fenchel conjugacy to the terms $\mathcal{D}_{U}$, is not any more linear, as in OT or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a nonlinear subhedging value. Our theory allows us to establish a nonlinear robust pricing-hedging duality, which covers a wide range of known robust results. We also focus on Wasserstein-induced penalizations and we study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.