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.
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $π_1, π_2, \ldots$ converges weakly to a transport plan $π$, then $π$ is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since opti…
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges …
The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as i…
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimension $d$ is arbitrary.
We explore the structure of solutions to a family of non-linear martingale optimal transport (MOT) problems that involve conditional expectations in the objective functional. En route general results concerning optimization over (martingale) measures are proved that appear much more widely applicable. In particular the analysis leads us to introduce a notion of so-called curtain transports; in a main contribution we highlight the rôle that these transports play in (non-linear) MOT.
Martingale transport plans on the line are known from Beiglbock & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d larger than one. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce…
We show continuity of the martingale optimal transport optimisation problem as a functional of its marginals. This is achieved via an estimate on the projection in the nested/causal Wasserstein distance of an arbitrary coupling on to the set of martingale couplings with the same marginals. As a corollary we obtain an independent proof of sufficiency of the monotonicity principle established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab., 44 (2016), no. 1, 42106] for cost functions of polynomial growth.
Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [21]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck, Jourdain, Margheriti, and Pammer [5], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w…
We study a variant of the martingale optimal transport problem in a multi-period setting to derive robust price bounds of a financial derivative. On top of marginal and martingale constraints, we introduce a time-homogeneity assumption, which restricts the variability of the forward-looking transitions of the martingale across time. We provide a dual formulation in terms of superhedging and discuss relaxations of the time-homogeneity assumption by adding market frictions. In financial terms, the introduced time-homogeneity corresponds to a time-consistency condition for call prices, given the …
We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we check that this algorithm performs better through numerical experiments. We also give a simple way to deal with the absence of convex ordering among the marginals. Furthermore, we provide a new universal bound on the error linked to entropy.
Consider a multiperiod optimal transport problem where distributions $μ_{0},\dots,μ_{n}$ are prescribed and a transport corresponds to a scalar martingale $X$ with marginals $X_{t}\simμ_{t}$. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence--Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if $μ_{0}$ is atomless, but not in general. In the one-period case …
We study the influence of additional intermediate marginal distributions on the value of the martingale optimal transport problem. From a financial point of view, this corresponds to taking into account call option prices not only, as usual, for those call options where the respective future maturities coincide with the maturities of some exotic derivative but also additional maturities and then to study the effect on model-independent price bounds for the exotic derivative. We characterize market settings, i.e., combinations of the payoff of exotic derivatives, call option prices and marginal…
Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglböck, Nutz & Touzi. Similar, we also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. As another contribution, we extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, and…
Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in general frameworks such as the one of Polish spaces. However, for the martingale transport problem several works based on different strategies established stability results for R only. We show that the restriction to dimension d = 1 is not accidental by presenting a sequence of marginal distributions on R 2 for which the martingale optimal transport problem is n…
We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions fo…
We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be approximated using linear programming (LP) problems which result from a discretisation of the marginal distributions combined with a suitable relaxation of the martingale constraint. Specialising to dimension one, we provide bounds on the convergence rate of the above scheme. We also show a stability result under only partial specification of the marginal dist…
We introduce a Benamou-Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous med…
Given a family of real probability measures $(μ_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(ν^α)_{α\in [0,1]}$ with $ν^α(\mathbb{R})=α$ that are submeasures of $μ_0$, called a parametrization of $μ_0$. Then, for any $α$ we define an evolution $(η^α_t)_{t\geq 0}$ of the …
This paper analyzes the support of the conditional distribution of optimal martingale transport plans in higher dimension. In the context of a distance coupling in dimension larger than 2, previous results established by Ghoussoub, Kim & Lim show that this conditional transport is concentrated on its own Choquet boundary. Moreover, when the target measure is atomic, they prove that the support is concentrated on d+1 points, and conjecture that this result is valid for arbitrary target measure. We provide a structure result of the support of the conditional optimal transport for general Lipschi…
Convergence of Sinkhorn's Algorithm for Entropic Martingale Optimal Transport Problem
Università degli Studi di Padova（帕多瓦大学）；Université Évry Paris-Saclay（巴黎萨克雷大学埃夫里分校）；Université Paris-Dauphine, PSL（巴黎第九大学，PSL）；Shanghai Jiao Tong University（上海交通大学）
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem w…
We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal distributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $O\left( n^{-1/2}\right)$ error where $n$ is the number of samples of each of the individual mar…
We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $γ$ is optimal if and only if there is a maximal monotone set $G$ that supports the $x$-marginal of $γ$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the support of $γ$. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading f…
