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

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.

Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22(μ,ν) between two probability measures μ and ν with finite second order moments on ℝd is the composition of a martingale coupling with an optimal transport map 𝛵. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and 𝛵#μ. Next, we give a direct proof that σ ↦ W22(σ,ν) is differentiable at μ in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function F on L2(Ω, ℙ; ℝd) is enough for the Fréchet differential at X to be a measurable function of X.