Approximation of martingale couplings on the line in the weak adapted topology

TL;DR

This paper proves the stability of martingale couplings under small changes in marginals, with implications for numerical methods, optimal transport, and financial modeling of option prices.

Contribution

It establishes the stability of martingale couplings in the weak adapted topology, connecting small marginal variations to small changes in couplings, with applications in finance and optimal transport.

Findings

Stability of martingale couplings under marginal approximation

Continuity of robust option pricing with respect to market data

New proof of the monotonicity principle in martingale optimal transport

Abstract

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.