In this paper, we exhibit a new family of martingale couplings between two onedimensional probability measures µ and ν in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of µ and ν. It contains the inverse transform martingale coupling which is explicit in terms of the quantile functions of these marginal densities.The integral of |x − y| with respect to each of these couplings is smaller than twice the W1 distance between µ and ν. When the comonotonous coupling between µ and ν is given by a map T , the elements of the family minimise R |y − T (x)| M (dx, dy) among all martingale couplings between µ and ν. When µ and ν are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.

Our main result is to establish stability of martingale couplings: suppose that π is a martingale coupling with marginals µ, ν. Then, given approximating marginal measures μ ≈ µ, ν ≈ ν in convex order, we show that there exists an approximating martingale coupling π ≈ π with marginals μ, ν.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.

Our main result is to establish stability of martingale couplings: suppose that $$\pi $$ π is a martingale coupling with marginals $$\mu , \nu $$ μ , ν . Then, given approximating marginal measures $$\tilde{\mu }\approx \mu , \tilde{\nu }\approx \nu $$ μ ~ ≈ μ , ν ~ ≈ ν in convex order, we show that there exists an approximating martingale coupling $$\tilde{\pi }\approx \pi $$ π ~ ≈ π with marginals $$\tilde{\mu }, \tilde{\nu }$$ μ ~ , ν ~ . 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 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.