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

Abstract

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.

Stability of the Weak Martingale Optimal Transport Problem

“…In the companion paper [10] we prove that any coupling whose marginals are approximated by probability measures can be approximated by couplings with respect to the adapted Wasserstein distance (see Proposition 2.4 below).…”

Instability of Martingale optimal transport in dimension d $\ge$ 2

“…The general version of (S1) and (S2) was first shown by Backhoff-Veraguas and Pammer [2, Theorem 1.1, Corollary 1.2] and Wiesel [19,Theorem 2.9]. Only very recently, Beiglböck, Jourdain, Margheriti and Pammer [4] have proven (S3). We want to stress that (S1), (S2) and (S3) are given in a minimal formulation and that in the articles some aspects of the results are notably stronger.…”

“…Moreover, it is an important achievement that on top of weak convergence we have convergence w.r.t. (an extension of) the adapted Wasserstein metric for the approximation in (S3) [4] and for the convergence in (S1) [5], see also [19]. Finally, these stability results also hold for weak martingale optimal transport which is an extension of (MOT) w.r.t.…”

On the stability of the martingale optimal transport problem: A set-valued map approach

“…(ii) For any x ∈ X, if x (n) → x for n → ∞, then for each y ∈ ϕ(x) there exists a subsequence x (n k ) k∈N and elements y (k) ∈ ϕ x (n k ) for each k ∈ N such that y (k) → y for k → ∞. The following result from [4] turns out to be crucial for our arguments. Note that the adapted 1-Wasserstein distance between two measures Q, Q ′ (with first marginals µ 1 and µ ′ 1 , respectively) is defined as…”

“…The stability result was indeed established for a Lipschitz-continuous payoff function and a relaxed formulation of the transport problem in [9], for particular payoffs and special marginals in [14], and eventually in a great generality by [2] and [19]. Recently, [4] established a result which allows to approximate martingale measures with fixed marginals (µ 1 , µ 2 ) in the adapted Wasserstein-distance (compare [2]) by a sequence of approximating martingale measures with fixed marginals (µ…”