Geometry of Martingale Optimal Transport and the Cost of Robust Bounds

Hiew, Lim, Pass, and Cruz de Souza investigate the geometry of optimal solutions to the martingale optimal transport problem, proving that for a broad class of payoff functions Φ(x, y) — including all convex-concave payoffs and many exotic option payoffs — the optimizer Q* ∈ M(μ, ν) is supported on a low-dimensional set in the product space R × R, specifically on the graph of a measurable function or on a set of dimension at most one (a curve) in each fiber {x} × R. This concentration of the optimizer's support — called the martingale transport map or the irreducible martingale transport — provides two simultaneous benefits: it gives structural insight into what the model-free worst-case dynamics look like (they are supported on a degenerate coupling rather than a diffuse distribution over all of R²), and it provides a route to tractable numerical computation of the bounds since the optimization can be restricted to the low-dimensional support class rather than over all of M(μ, ν). The dimension-reduction result exploits the geometry of the convex order condition and the martingale constraint to show that the optimizer must satisfy a necessary condition — the co-monotonicity or left-curtain property — that forces the support of Q* to be contained in a specific one-dimensional set determined by the payoff Φ and the marginals (μ, ν). For the class of payoffs that are convex in y — including calls, puts, and variance swap payoffs — the optimizer of the upper bound UB = sup_{Q ∈ M(μ,ν)} E_Q[Φ(X,Y)] is the Kellerer left-curtain coupling, which transports each atom of μ to a two-point distribution supported on adjacent quantile levels of ν; for payoffs convex in x and concave in y (including some corridor options and spreads), a different degenerate coupling structure arises. The key insight is that the optimizer is always extremal in the convex set M(μ,ν) — it lies on the boundary rather than the interior — and extreme points of M(μ,ν) have the low-dimensional support property by the Choquet representation theorem applied to the compact convex set M(μ,ν). The computational consequence is that the numerical MOT problem can be solved on the low-dimensional support rather than by discretizing the full product measure on a grid, reducing the problem from O(n²) variables (a coupling over n × n grid points) to O(n) variables (a one-dimensional transport on the support manifold), enabling exact computation of model-free bounds for realistic option-market grids without the scaling limitations that make the full MOT problem computationally expensive.

The upper martingale transport bound is UB = sup_{Q ∈ M(μ,ν)} E_Q[Φ(X,Y)] where M(μ,ν) = {Q ∈ P(R²) : Q_1 = μ, Q_2 = ν, E_Q[Y | X] = X} is the set of martingale couplings. The left-curtain coupling π^{lc}_{μ,ν} is the unique martingale coupling with the left-monotone property: there exist functions T_l(x) ≤ x ≤ T_r(x) such that Q_{lc}({(x,y) : y ∈ {T_l(x), T_r(x)}}) = 1 and E_Q[Y | X = x] = x, so that each atom of μ at x is transported to a convex combination of exactly two points T_l(x) and T_r(x) in ν. The left-curtain attains UB for all convex Φ(x, y) by the Hobson–Neuberger theorem. For general Φ, the optimizer Q* satisfies the necessary condition that Q* is concentrated on the set {(x,y) : y ∈ ∂_y Ψ(x)} where Ψ(x) = ∫ Φ(x, y) Q*_x(dy) is the fiberwise expected payoff and ∂_y Ψ denotes the superdifferential with respect to y, a condition on the support of Q* that generically restricts it to a one-dimensional manifold in R². The dimension of the support is measured by the Hausdorff dimension of the projection of supp(Q*) onto R, which is at most one for all payoffs satisfying the appropriate subdifferential conditions.

The left-curtain property — transport of each atom of μ to a two-point distribution in ν — is geometrically interpretable as the martingale analogue of the classical monotone rearrangement: in classical optimal transport, the optimizer for convex costs is the unique coupling where X and Y are co-monotone (X ↑ as Y ↑); in martingale transport, the optimizer for convex payoffs is the unique coupling where each X is transported to a two-point distribution immediately to its left and right in ν, the minimal way to spread mass subject to the martingale constraint while keeping X in the convex hull of the transported ν-mass. The uniqueness of the left-curtain for convex payoffs is the martingale analogue of the classical Brenier theorem, and its constructive characterization via the functions T_l and T_r enables efficient numerical computation. The Choquet representation theorem argument — that extreme points of the compact convex set M(μ,ν) have the low-dimensional support property — is the structural explanation for why the optimizer is always concentrated on a low-dimensional set: the convex set M(μ,ν) is a subset of the space of probability measures on R², and its extreme points are the atomic measures supported on sets of minimal dimension consistent with the marginal and martingale constraints. By the Choquet theorem, any point in M(μ,ν) is a convex combination of extreme points, and the optimal measure Q* lies on the boundary face of M(μ,ν) that is exposed by the linear functional Q → E_Q[Φ], placing it at or near the extreme points of M(μ,ν) and hence on low-dimensional support. The extension to the multi-period case — n-marginal martingale transport for n ≥ 3 — is substantially harder: the low-dimensional support property extends only partially, as the optimizer of the n-period problem can be supported on a set of dimension up to n − 1 in R^n rather than dimension 1 in R², and the Kellerer-type characterization requires solving an iterative sequence of two-period problems that may not have closed-form solutions. For n ≥ 3 the computational savings from dimension reduction are less dramatic, and alternative methods such as the Sinkhorn algorithm for entropic regularization are used in practice.

The computational savings from the left-curtain characterization make martingale transport bounds practically computable for the desk on a daily calibration basis: instead of solving a linear programme on a n × n grid of (μ, ν) discretization points — with n typically 50 to 100 strikes per maturity slice — the left-curtain can be computed by finding T_l and T_r via a sorting algorithm on the marginals, reducing the computational cost from O(n³) for a general LP (via simplex) to O(n log n) for the sorting. This makes model-free bounds computable in real time for any bi-marginal payoff, enabling the desk to monitor bounds continuously across the trading day as the vanilla surface updates. For corridor variance swaps and other payoffs with complex dependence on both marginals, the dimension-reduction result provides a classification of which payoffs admit tractable left-curtain-type optimizers (convex payoffs in Y), which require a different but equally tractable characterization (concave-convex payoffs), and which require a full LP solve (general mixed payoffs). This classification is immediately useful for the desk's exotic book: for payoffs in the first class, the optimal bound computation is O(n log n); for the second, a similar O(n log n) algorithm applies; and for the third, full LP computation at O(n³) cost is required, signalling that the bound computation deserves more careful attention to numerical precision and convexification before reporting.

The low-dimensional support property holds for payoffs satisfying specific convexity or concavity conditions that characterize the payoffs for which the optimal coupling is extremal in M(μ,ν); for payoffs that are neither convex nor concave in Y — including some exotic structured products with complex path-dependent features or multi-threshold dependencies — the optimizer may not be the left-curtain or any other easily characterized coupling, and the support dimension may approach two (the full product space R²), making the computational savings from dimension reduction unavailable. The desk must verify the payoff convexity structure before applying the left-curtain algorithm and revert to full LP computation for non-convex payoffs. The extension to continuous-time marginals — where μ and ν are the risk-neutral marginals at maturities T_1 and T_2 implied by vanilla option prices rather than discrete probability distributions on a grid — requires passing from the discrete MOT problem to a continuum version whose optimizer is characterized by a transport map between measure-valued marginals. The existence theory for the continuum MOT problem is established (Kellerer 1972, Beiglböck et al. 2013), but the numerical solution requires discretization of the marginals on a fine grid, introducing a discretization error in the bound that depends on the grid resolution and the smoothness of the marginals. For marginals with heavy tails (typical for equity options), the grid must extend to extreme strikes at which the implied distribution has very low probability, requiring many grid points to achieve adequate tail coverage and making the discretization of μ and ν expensive.