Geometry of vectorial martingale optimal transportations and duality

TL;DR

This paper introduces the Vectorial Martingale Optimal Transport (VMOT) problem, extending classical MOT to multiple assets, and proves the existence of dual optimizers that inform optimal hedging strategies and the geometry of solutions.

Contribution

It formulates VMOT as an infinite-dimensional linear program and establishes the existence of dual optimizers, revealing geometric properties of optimal martingale couplings in multi-asset settings.

Findings

Dual optimizers exist for the VMOT problem.

Optimal martingales exhibit extremal correlation structures.

Results inform robust financial hedging strategies.

Abstract

The theory of Optimal Transport (OT) and Martingale Optimal Transport (MOT) were inspired by problems in economics and finance and have flourished over the past decades, making significant advances in theory and practice. MOT considers the problem of pricing and hedging of a financial instrument, referred to as an option, assuming its payoff depends on a single asset price. In this paper we introduce Vectorial Martingale Optimal Transport (VMOT) problem, which considers the more general and realistic situation in which the option payoff depends on multiple asset prices. We address this problem of pricing and hedging given market information -- described by vectorial marginal distributions of underlying asset prices -- which is an intimately relevant setup in the robust financial framework. We establish that the VMOT problem, as an infinite-dimensional linear programming, admits an optimizer for its dual program. Such existence result of dual optimizers is significant for several reasons: the dual optimizers describe how a person who is liable for an option payoff can formulate optimal hedging portfolios, and more importantly, they can provide crucial information on the geometry of primal optimizers, i.e. the VMOTs. As an illustration, we show that multiple martingales given marginals must exhibit an extremal conditional correlation structure whenever they jointly optimize the expectation of distance-type cost functions.