  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.
