  While many questions in robust finance can be posed in the martingale optimal
transport framework or its weak extension, others like the subreplication price
of VIX futures, the robust pricing of American options or the construction of
shadow couplings necessitate additional information to be incorporated into the
optimization problem beyond that of the underlying asset. In the present paper,
we take into account this extra information by introducing an additional
parameter to the weak martingale optimal transport problem. We prove the
stability of the resulting problem with respect to the risk neutral marginal
distributions of the underlying asset, thus extending the results in
\cite{BeJoMaPa21b}. A key step is the generalization of the main result in
\cite{BJMP22} to include the extra parameter into the setting. This result
establishes that any martingale coupling can be approximated by a sequence of
martingale couplings with specified marginals, provided that the marginals of
this sequence converge to those of the original coupling. Finally, we deduce
stability of the three previously mentioned motivating examples.
