Discrete Martingale Optimal Transport

• Discrete Martingale Optimal Transport is a discrete-time optimal transport framework that imposes a martingale constraint to ensure risk-neutral, mean-preserving transport between given marginals.
• It features a primal-dual structure where optimal coupling links directly with hedging strategies, using linear programming and entropic regularization methods under conditions like the Spence–Mirrlees criterion.
• The approach underpins robust pricing in multi-asset, multi-period finance, with applications extending from Asian option pricing to bi-martingale and information-based extensions.

Discrete martingale optimal transport (MOT) is the discrete-time optimal transport problem under the additional requirement that the transport plan be a martingale coupling. In its most standard one-step form, one fixes two probability measures $\mu,\nu$ on $\mathbb{R}$ or $\mathbb{R}^d$, searches over couplings $\pi$ with those marginals, and imposes $\mathbb{E}[Y\mid X]=X$. In the discrete setting this becomes a finite-dimensional linear optimization problem on a coupling matrix or tensor, and in mathematical finance it is the canonical formulation of model-independent pricing and robust hedging under marginal information extracted from option prices (Beiglböck et al., 2012, Bäuerle et al., 2019, Guo et al., 2017).

1. Formal setup and feasibility

For one-step discrete MOT with finitely supported marginals

$$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$

a martingale coupling is a nonnegative matrix $(\pi_{ij})$ satisfying the marginal constraints

$$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$

together with the row-wise martingale constraint

$\sum_j y_j \pi_{ij}=x_i\mu_i.$

Equivalently, if $\pi(dx,dy)=\mu(dx)\,\pi_x(dy)$, then $\mathbb{R}$0 $\mathbb{R}$1-a.s., or $\mathbb{R}$2 (Backhoff-Veraguas et al., 2019).

The corresponding primal optimization problem is

$\mathbb{R}$3

or, depending on the application, the supremum version

$\mathbb{R}$4

In multi-period discrete time, one fixes a horizon $\mathbb{R}$5, a canonical process $\mathbb{R}$6, and the martingale class

$\mathbb{R}$7

then intersects it with the set of laws having prescribed one-dimensional marginals $\mathbb{R}$8 (Zhou et al., 2021).

Feasibility is governed by convex order. In the one-step case, Strassen’s theorem gives

$\mathbb{R}$9

meaning that $\mathbb{R}^d$0 and $\mathbb{R}^d$1 have the same mass and barycenter and satisfy $\mathbb{R}^d$2 for every convex $\mathbb{R}^d$3 (Backhoff-Veraguas et al., 2019, Beiglböck et al., 2012). In multi-period discrete MOT, the analogous consistency requirement is a convex-order chain $\mathbb{R}^d$4 (Zhou et al., 2021). For multi-asset vectorial MOT, the data may consist only of one-dimensional marginals $\mathbb{R}^d$5 for each asset $\mathbb{R}^d$6 and time $\mathbb{R}^d$7, with componentwise convex-order constraints $\mathbb{R}^d$8 (Lim, 2016, Che et al., 3 Feb 2026).

2. Primal–dual structure and robust hedging

The discrete MOT primal has a direct financial dual. In a standard multi-period discrete formulation, for a path-dependent payoff $\mathbb{R}^d$9, the dual variables are dynamic trading strategies $\pi$0 and static terminal payoffs $\pi$1, with pathwise superhedging constraint

$\pi$2

The dual value is the minimal cost of such a semi-static superhedge, typically written as an infimum over $\pi$3, and the discrete MOT duality theorem identifies this value with the supremum of $\pi$4 over martingale measures with the prescribed marginals (Dolinsky et al., 2012).

On the line, a general duality theory requires some care. The classical pointwise dual constraint

$\pi$5

is not sufficient in full generality: there may be a duality gap even for bounded lower semicontinuous $\pi$6, and there may be no dual optimizer even for continuous $\pi$7 and compactly supported marginals. A quasi-sure formulation, based on $\pi$8-polar sets and irreducible convex-order components, yields complete duality for Borel $\pi$9: no duality gap and existence of dual optimizers (Beiglböck et al., 2015). This corrected duality framework is central whenever discrete MOT is used as an approximation of continuous or irregular problems.

The robust-hedging interpretation is not merely formal. In model-independent finance, call prices determine the marginals, martingale couplings represent all risk-neutral models consistent with those marginals, and the primal MOT value is the extremal model-consistent price of the exotic payoff. The dual is the cheapest semi-static strategy that super-replicates the payoff pathwise (Dolinsky et al., 2012, Bäuerle et al., 2019). A common misconception is that discrete MOT is only a combinatorial transport problem; the duality theory shows that it is also a superhedging theory.

3. Geometry of optimal couplings

A defining feature of MOT is that optimal plans are governed by a martingale analogue of cyclical monotonicity. Beiglböck and Juillet established a variational principle: if $\mathbb{E}[Y\mid X]=X$0 is optimal, then there exists a Borel set $\mathbb{E}[Y\mid X]=X$1 with $\mathbb{E}[Y\mid X]=X$2 such that every finite measure $\mathbb{E}[Y\mid X]=X$3 supported on $\mathbb{E}[Y\mid X]=X$4 has smaller cost than any competitor $\mathbb{E}[Y\mid X]=X$5 with the same marginals and the same conditional barycenters. This finitistic local optimality replaces the support-based $\mathbb{E}[Y\mid X]=X$6-cyclical monotonicity of classical OT (Beiglböck et al., 2012).

In one dimension, the canonical optimal martingale coupling is the left-curtain, or left-monotone, coupling. It is characterized by the exclusion of configurations $\mathbb{E}[Y\mid X]=X$7 with $\mathbb{E}[Y\mid X]=X$8 and $\mathbb{E}[Y\mid X]=X$9. When $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$0 is continuous, the left-curtain coupling is supported on the graphs of two functions $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$1, with $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$2. For costs of the form $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$3 with strictly convex derivative $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$4, this coupling is the unique optimizer (Beiglböck et al., 2012). In the purely discrete case, Bäuerle and Schmithals proved the optimality of left-monotone transport plans under a martingale Spence–Mirrlees condition and gave an explicit finite algorithm for constructing the unique left-monotone plan (Bäuerle et al., 2019).

Special costs can induce more refined support geometry. For the discrete Asian-option reduction with cost $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$5, maximizers can be chosen supported on the union of the graphs of two measurable functions $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$6, with $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$7 non-increasing, while minimizers can be chosen supported on two graphs together with the secondary diagonal $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$8, with the upper branch non-decreasing (Stebegg, 2014). This is a concrete instance in which the geometry of the optimal discrete martingale transport can be read off directly from the payoff structure.

In vectorial martingale optimal transport, where the payoff depends on several asset prices and only one-dimensional marginals are prescribed, optimal conditional laws exhibit extremal geometry. For distance-type costs $$\mu=\sum_{i=1}^I \mu_i \delta_{x_i},\qquad \nu=\sum_{j=1}^J \nu_j \delta_{y_j},$$9 under strictly convex norms, the support of the conditional law $(\pi_{ij})$0 consists of extreme points of its convex hull, and for the minimizing distance cost, any mass common to both marginals stays on the diagonal (Lim, 2016). A plausible implication is that, even in finite-state multi-asset discretizations, extremality and “staying” phenomena can sharply restrict candidate supports.

4. Stability, approximation, and limits

Stability is a central issue because empirical or numerical marginals are noisy. For one-step MOT on $(\pi_{ij})$1, if $(\pi_{ij})$2 uniformly, $(\pi_{ij})$3 and $(\pi_{ij})$4 in $(\pi_{ij})$5, and $(\pi_{ij})$6 are optimizers for $(\pi_{ij})$7, then any weak accumulation point of $(\pi_{ij})$8 is an optimizer for $(\pi_{ij})$9; if the limit optimizer is unique, then $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$0. The proof uses an adapted weak topology that tracks the disintegrations $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$1 and a notion of martingale $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$2-monotonicity, since ordinary support monotonicity is not stable enough in the martingale setting (Backhoff-Veraguas et al., 2019).

For computational approximation of continuous MOT by discrete LPs, a standard device is to discretize marginals and relax the martingale constraint. Guo and Obłój define $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$3-approximating martingale measures and prove that LP values based on discrete marginals $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$4 and martingale relaxation $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$5 converge to the true MOT value as soon as the Wasserstein discretization error is dominated by $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$6. In the one-step, one-dimensional case, if $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$7 has bounded support, the approximation error is bounded linearly in $$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$8 (Guo et al., 2017). They also describe a one-dimensional discretization scheme, due to Dolinsky and Soner, that preserves convex order and therefore avoids martingale relaxation in that setting (Guo et al., 2017).

Distributional robustness pushes this further. The distributionally robust MOT problem replaces exact marginals by a Wasserstein neighborhood of benchmark marginals and studies

$$\sum_j \pi_{ij}=\mu_i,\qquad \sum_i \pi_{ij}=\nu_j,$$9

This relaxed problem admits strong duality and a finite-dimensional LP approximation, and its empirical version can be approximated within $\sum_j y_j \pi_{ij}=x_i\mu_i.$0 error in the one-dimensional setting, where $\sum_j y_j \pi_{ij}=x_i\mu_i.$1 is the number of samples of each individual marginal distribution; in $\sum_j y_j \pi_{ij}=x_i\mu_i.$2-dimensional marginals the rate becomes $\sum_j y_j \pi_{ij}=x_i\mu_i.$3 (Zhou et al., 2021). This establishes a direct bridge between discrete MOT, data-driven uncertainty sets, and non-asymptotic sample complexity.

The relation between discrete and continuous MOT is also structural. Dolinsky and Soner’s continuous-time robust hedging result constructs simple piecewise constant super-replication portfolios and proves $\sum_j y_j \pi_{ij}=x_i\mu_i.$4, showing that discrete-time approximations recover the continuous-time martingale transport duality (Dolinsky et al., 2012). Huesmann and Trevisan’s Benamou–Brenier formulation interprets continuous-time MOT as a weak length relaxation of discrete-time MOT, with a corresponding PDE/Fokker–Planck reformulation (Huesmann et al., 2017). This suggests that discrete MOT is not merely a finite approximation device; it is also the finite-step manifestation of a broader martingale transport geometry.

5. Computational methods

Because discrete MOT is a finite-dimensional linear program, LP methods remain the baseline computational approach. In the one-step discrete case, the primal variables are the coupling entries $\sum_j y_j \pi_{ij}=x_i\mu_i.$5, and the dual variables $\sum_j y_j \pi_{ij}=x_i\mu_i.$6 encode a static–dynamic superhedge. Under the martingale Spence–Mirrlees condition, the support structure becomes sufficiently rigid that a constructive algorithm for the unique left-monotone optimizer is available (Bäuerle et al., 2019).

General discretization-based LP schemes were systematized by Guo and Obłój. Their framework covers deterministic and random discretizations, relaxed martingale constraints, convergence guarantees, and a convex-order-preserving discretization in one dimension. They also discuss entropic regularization and iterative Bregman projection as natural computational extensions of the LP formulation (Guo et al., 2017).

Recent work has made the entropic approach explicit for discrete MOT. In finite-support problems, the martingale condition is the row-wise equality $\sum_j y_j \pi_{ij}=x_i\mu_i.$7, and a supermartingale condition becomes $\sum_j y_j \pi_{ij}=x_i\mu_i.$8. Since exact martingale constraints are typically infeasible after discretization, the problem is relaxed to

$\sum_j y_j \pi_{ij}=x_i\mu_i.$9

An entropic formulation then leads to Sinkhorn-type algorithms with sparse Newton iterations. The proposed methods are designed for discrete MOT and more general OT problems with row-wise structural constraints; in practice they enjoy super-exponential convergence and robustness with controllable thresholds for total constraint violations (Tang et al., 25 Aug 2025).

Large-scale multi-period multi-asset discrete MOT can also be handled by first-order primal–dual methods. A recent study on dual attainment in multi-period multi-asset MOT uses a discrete LP approximation together with the Primal-Dual Linear Programming (PDLP) algorithm and reports practical solvability of large-scale instances arising from robust pricing of worst-of autocallable options (Che et al., 3 Feb 2026). This indicates that discrete MOT computation is no longer confined to low-dimensional toy problems, although a plausible implication is that the effective complexity still depends strongly on sparsity, factorization, and problem geometry.

6. Applications, extensions, and current directions

The most developed application remains model-independent finance. Discrete Asian-option pricing with two monitoring dates reduces to a two-step MOT problem with cost $\pi(dx,dy)=\mu(dx)\,\pi_x(dy)$0, and the maximizing and minimizing models admit explicit geometric descriptions in terms of optimal martingale transports (Stebegg, 2014). More broadly, robust bounds for path-dependent derivatives are expressed as discrete or continuous MOT problems, and the dual variables describe semi-static hedging portfolios (Dolinsky et al., 2012).

Multi-asset and multi-period extensions enlarge the admissible data and payoff classes. Vectorial MOT considers payoffs $\pi(dx,dy)=\mu(dx)\,\pi_x(dy)$1 depending on several assets while only one-dimensional marginals are prescribed; the dual problem then uses static positions in one-dimensional marginals and dynamic trading in all assets (Lim, 2016). A recent generalization establishes dual attainment for multimarginal, multi-asset MOT under mild regularity and irreducibility conditions and supports this theory with discrete numerical experiments for worst-of autocallables (Che et al., 3 Feb 2026).

Several newer variants modify the classical MOT objective rather than the admissible class. Information-based MOT introduces randomized arcade processes and filtered arcade martingales, then optimizes over martingale couplings by maximizing a functional built from the pathwise filtering error of the resulting continuous-time martingale. In the one-step case, existence and uniqueness of the optimal coupling are established, and an algorithm for empirical measures is proposed (Kassis et al., 2024). This places discrete MOT in an information-theoretic setting in which the coupling is chosen as the worst martingale model for a prescribed noisy information flow.

Another extension addresses the instability of MOT under marginal perturbations in higher dimensions. Bi-martingale optimal transport introduces two martingale-type constraints and an intermediate measure dominating both marginals in convex order. For quadratic cost, it yields an optimal transport interpretation of the second Zolotarev distance; more importantly for discrete MOT, it provides a $\pi(dx,dy)=\mu(dx)\,\pi_x(dy)$2-convergent bi-martingale approximation of classical MOT that robustly accommodates deviations from convex order and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions (Bołbotowski, 31 Oct 2025).

Two recurrent misconceptions are therefore inaccurate. First, exact discrete MOT is not intrinsically stable under arbitrary marginal perturbations; stability may require adapted topologies, convex-order-preserving discretizations, or explicit regularization (Backhoff-Veraguas et al., 2019, Bołbotowski, 31 Oct 2025). Second, discrete MOT is not merely a simplified special case of a continuous theory; it is simultaneously a financially interpretable robust-hedging problem, a standalone LP or conic optimization problem, and a mathematically meaningful approximation regime for continuous-time martingale transport (Dolinsky et al., 2012, Huesmann et al., 2017).