MARGHERITI William

We are interested in martingale rearrangement couplings. As introduced by Wiesel [37] in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Fr\'echet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in [24] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fr\'echet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglb\"ock and Juillet [9] and which involve the uniform distribution on [0, 1] in addition to the two marginals. We last discuss the stability in adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginal distributions.

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.

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.

It is known since [24] that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of $\vert x-y\vert$ is smaller than twice their $\mathcal W_1$-distance (Wasserstein distance with index $1$). We showed in [24] that replacing $\vert x-y\vert$ and $\mathcal W_1$ respectively with $\vert x-y\vert^\rho$ and $\mathcal W_\rho^\rho$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing $\mathcal W_\rho^\rho$ with the product of $\mathcal W_\rho$ times the centred $\rho$-th moment of the second marginal to the power $\rho-1$. Then we study the generalisation of this new stability inequality to higher dimension.

This thesis is motivated by the study of the stability of the martingale optimal transport problem, and is naturally organized into two parts. In the first part, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν comparable in convex order. In particular, this family contains the inverse transform martingale coupling, which is explicit in terms of quantile functions of the marginals. The integral M_1(μ,ν) of |x-y| against each of these couplings is increased by twice the Wasserstein distance W_1(μ,ν) between μ and ν. We show a similar inequality when |x-y| and W_1 are respectively replaced by |x-y|^ρ and the product of W_ρ by the centered moment of order ρ of the second marginal raised to the exponent ρ-1, for any ρ∈[1,+∞[. We then study the generalization of this new stability inequality to the higher dimension. Finally, we establish a strong connection between our new family of martingale couplings and the projection of a coupling between two comparable given marginals in convex order onto the set of martingale couplings between these same marginals. This last projection is taken with respect to the adapted Wasserstein distance, which majors the usual Wasserstein distance and thus induces a finer topology better suited for financial modeling, since it takes into account the temporal structure of martingales. In the second part, we prove that any martingale coupling whose marginals are approximated by comparable probability measures in convex order can itself be approximated by martingale couplings in the sense of the adapted Wasserstein distance. We then discuss various applications of this result. In particular, we strengthen a stability result for the weak optimal transport problem and establish a stability result for the weak optimal martingale transport problem.

This thesis is motivated by the study of the stability of the martingale optimal transport problem, and is naturally organized into two parts. In the first part, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν comparable in convex order. In particular, this family contains the inverse transform martingale coupling, which is explicit in terms of quantile functions of the marginals. The integral M_1(μ,ν) of |x-y| against each of these couplings is increased by twice the Wasserstein distance W_1(μ,ν) between μ and ν. We show a similar inequality when |x-y| and W_1 are respectively replaced by |x-y|^ρ and the product of W_ρ by the centered moment of order ρ of the second marginal raised to the exponent ρ-1, for any ρ∈[1,+∞[. We then study the generalization of this new stability inequality to the higher dimension. Finally, we establish a strong connection between our new family of martingale couplings and the projection of a coupling between two comparable given marginals in convex order onto the set of martingale couplings between these same marginals. This last projection is taken with respect to the adapted Wasserstein distance, which majors the usual Wasserstein distance and thus induces a finer topology better suited for financial modeling, since it takes into account the temporal structure of martingales. In the second part, we prove that any martingale coupling whose marginals are approximated by comparable probability measures in convex order can itself be approximated by martingale couplings in the sense of the adapted Wasserstein distance. We then discuss various applications of this result. In particular, we strengthen a stability result for the weak optimal transport problem and establish a stability result for the weak optimal martingale transport problem. We derive stability with respect to the marginals of the over-replication price of VIX futures contracts.