When approximating in Wasserstein distance the

two marginals of a martingale coupling on the

real by probability measures in the convex order,

it is possible to construct a sequence of

martingale couplings between these probability

measures converging in adapted Wasserstein

distance to the original coupling. We deduce the

stability with respect to the marginal

distributions of the Weak Martingale Optimal

Transport problem in dimension one. As an

application, we obtain the stability of the

superreplication price of the VIX future. To deal

with the subreplication price, we need a

generalisation of our main result to extended

martingale couplings with an extra parameter

which enables to take into account additional information.