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

G. Pammer,M. Beiglböck,W. Margheriti,B. Jourdain

doi:10.1007/s00440-021-01103-y

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 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.

Highlights

Before carefully explaining all required notation and describing relevant literature, let us give a first description of our main result and its relevance for the martingale transport theory.While classical transport theory is concerned with the set (μ, ν) of couplings or transport plans of probability measures μ, ν, the martingale variant restricts the problem to the set M (μ, ν) of martingale couplings, that is, transport plans which preserve the barycenter of each particle
According to Strassen’s theorem [52], the existence of a martingale coupling between two probability measures μ, ν ∈ P(R) with finite first moment is equivalent to μ ≤c ν, where ≤c denotes the convex order
Once the Martingale Optimal Transport (MOT) problem is discretised by approximating μ and ν by probability measures with finite support and in the convex order, Alfonsi, Corbetta and Jourdain [3] raised the question of the convergence of optimal costs of the discretised problem towards the costs of the original problem

Summary

Introduction

Before carefully explaining all required notation and describing relevant literature, let us give a first description of our main result and its relevance for the martingale transport theory. This result is so basic and straightforward that its implicit use is overlooked Note that it plays a crucial role in a number of occasions, e.g. for stability of optimal transport, providing numerical approximations, or in the characterisation of optimality through cyclical monotonicity. The main result of this article is to establish Fact 1.1 for martingale transports on the real line, see Theorem 2.6 below This closes a gap in the theory of martingale transport and yields basic fundamental results in a unified fashion that is much closer to the classical theory. It allows to address questions in martingale optimal transport, optimal Skorokhod embedding and robust finance that have previously remained open.

The martingale optimal transport problem

The adapted Wasserstein distance

Outline

Main result

On the adapted weak topology

Auxiliary results on the convex order in dimension one

Proof of the main theorem