  Our main result is to establish stability of martingale couplings: suppose
that $\pi$ is a martingale coupling with marginals $\mu, \nu$. Then, given
approximating marginal measures $\tilde \mu \approx \mu, \tilde \nu\approx \nu$
in convex order, we show that there exists an approximating martingale coupling
$\tilde\pi \approx \pi$ with marginals $\tilde \mu, \tilde \nu$.
  In mathematical finance, prices of European call / put option yield
information on the marginal measures of the arbitrage free pricing measures.
The above result asserts that small variations of call / put prices lead only
to small variations on the level of arbitrage free pricing measures.
  While these facts have been anticipated for some time, the actual proof
requires somewhat intricate stability results for the adapted Wasserstein
distance. Notably the result has consequences for a several related problems.
Specifically, it is relevant for numerical approximations, it leads to a new
proof of the monotonicity principle of martingale optimal transport and it
implies stability of weak martingale optimal transport as well as optimal
Skorokhod embedding. On the mathematical finance side this yields continuity of
the robust pricing problem for exotic options and VIX options with respect to
market data. These applications will be detailed in two companion papers.
