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The theory of Optimal Transport (OT) and Martingale Optimal Transport (MOT)were inspired by problems in economics and finance and have flourished over thepast decades, making significant advances in theory and practice. MOT considersthe problem of pricing and hedging of a financial instrument, referred to as anoption, assuming its payoff depends on a single asset price. In this paper weintroduce Vectorial Martingale Optimal Transport (VMOT) problem, whichconsiders the more general and realistic situation in which the option payoffdepends on multiple asset prices. We address this problem of pricing andhedging given market information -- described by vectorial marginaldistributions of underlying asset prices -- which is an intimately relevantsetup in the robust financial framework.  We establish that the VMOT problem, as an infinite-dimensional linearprogramming, admits an optimizer for its dual program. Such existence result ofdual optimizers is significant for several reasons: the dual optimizersdescribe how a person who is liable for an option payoff can formulate optimalhedging portfolios, and more importantly, they can provide crucial informationon the geometry of primal optimizers, i.e. the VMOTs. As an illustration, weshow that multiple martingales given marginals must exhibit an extremalconditional correlation structure whenever they jointly optimize theexpectation of distance-type cost functions.

Geometry of vectorial martingale optimal transportations and duality