Martingale Optimal Transport

• Martingale optimal transport is a framework that imposes a martingale constraint on coupling problems, ensuring that the conditional expectation of the target equals the source for fair pricing.
• It employs duality and monotonicity principles, such as the left-curtain coupling, to achieve clear geometric characterizations and unique optimal solutions under the martingale Spence–Mirrlees condition.
• The theory extends to frictional settings with state-dependent costs, allowing for robust superhedging strategies and providing explicit analytical formulas in linear-quadratic cases.

Martingale optimal transport (MOT) is the study of optimal coupling problems under the constraint that the coupling must be a martingale. Originating in robust mathematical finance, MOT replaces the classical transport plan admissibility with the requirement that the conditional expectation of the target variable given the source equals the source itself, encoding no-arbitrage or fair-game dynamics in model-independent pricing frameworks. The theory extends from the frictionless setting—fully characterized in one dimension via left-monotone (left-curtain) couplings and robust duality—to MOT problems with convex state-dependent frictions, multi-step or continuous time settings, and numerically tractable relaxations.

1. Frictionless MOT: One-Step Structure and Duality

Let $\mu, \nu$ be Borel probability measures on $\mathbb{R}$ with finite first moments and $\mu \preceq_{cx} \nu$, i.e., $\int f\, d\mu \leq \int f\, d\nu$ for all convex $f$. Strassen’s theorem ensures the existence of at least one martingale coupling between $\mu$ and $\nu$, i.e., a joint law $\pi$ on $(X, Y)$ with $X \sim \mu$, $Y \sim \nu$ and $\mathbb{E}_\pi[Y \mid X] = X$ $\mu$-a.s.

The frictionless MOT problem is, for cost $\Phi(x, y)$ (measurable, integrable),

$$\sup_{\pi \in \Pi^M(\mu, \nu)} \mathbb{E}_\pi[\Phi(X, Y)],$$

where $\Pi^M(\mu, \nu)$ is the set of all martingale couplings. The dual formulation, established by Beiglböck–Juillet and Beiglböck–Nutz–Touzi, introduces triplets $(\varphi, \psi, h)$, with $\varphi, \psi$ functions and $h$ a predictable "direction," satisfying

$$\varphi(x) + \psi(y) + h(x)(y - x) \geq \Phi(x, y), \ \forall x, y.$$

The dual value is then

$$\inf_{(\varphi, \psi, h)} \mu(\varphi) + \nu(\psi).$$

Strong duality and attainment hold under mild regularity. For cost functions satisfying a martingale Spence–Mirrlees (MSM) condition (convexity in certain variables, e.g., $\partial_{xyy}\Phi > 0$), the optimizer is unique and supported on two monotone graphs $T_d(x) \leq x \leq T_u(x)$, where $T_d$ is nonincreasing and $T_u$ nondecreasing.

2. Geometry: Monotonicity, Left-Curtain Coupling, and Frictionless Optimal Structures

The "left-curtain" (left-monotone) coupling, introduced by Beiglböck–Juillet, provides a canonical construction of the optimizer in the case of one dimension and MSM costs. The support of the optimal plan forbids "crossings" or forbidden rectangles, leading to robust characterizations and explicit mass-splitting (typically into two points per $x$ in the atomless case). This structure heavily informs the analysis and algorithms for more complex settings.

In the multi-step setting, monotonicity principles hold recursively. The optimizer admits a bi-atomic disintegration per time step, with the plan constructed in terms of monotone graphs on active intervals. Mass-balance and equal-slope (touching tangent) conditions exclusively determine these graphs off the “band.”

3. Extension: Frictional MOT with State-Dependent Trading Costs

Introducing convex, state-dependent frictional trading costs $f_t(x, \delta) = a_t(x)|\delta| + b_t(x)\delta^2$, with $a_t, b_t \geq 0$ and $b_t > 0$ on compacts, fundamentally alters the optimization. The frictional primal considers

$$V_{\mathrm{fric}} = \sup_{\pi\in\mathcal{M}} \left\{\mathbb{E}_\pi[\Phi] - \sum_{t=0}^{N-1} \mathbb{E}_\pi[f_t(S_t, S_{t+1} - S_t)]\right\},$$

with $\mathcal{M}$ the set of multi-marginal martingale laws matching the prescribed marginals. The dual introduces a pathwise inequality for each $t$: $$\varphi_t(x) + \psi_t(y) + h_t(x)(y - x) \leq f_t(x, y-x)$$ and minimizes the total cost over all such collections, with attainment and no duality gap under reasonable convexity assumptions (Rai, 9 Oct 2025).

Friction dramatically changes the fine geometry. There exists a non-trivial “trade band,” defined by a subdifferential condition on the dual slope $h_t(x) \in \partial_v f_t(x, 0)$. Within this band, optimal transport is the identity; off the band, mass transports along two monotone graphs, and their locations are determined by an “equal-slope” system: $$\partial_y \widetilde{c}_t(x, T_d(x)) = \partial_y \widetilde{c}_t(x, T_u(x)) =: \alpha_t(x)$$ where $\widetilde{c}_t$ is the continuation-adjusted cost. The trade band automatically widens with increased linear friction and shrinks with increased quadratic friction; explicit formulas are available in the linear-quadratic case.

4. Duality, Dynamic Programming, and Stability

Frictional MOT preserves strong duality and dynamic programming structure, enabling recursive computation of the value function: $$\int V_t\,d\mu_t = \int V_{t+1}\,d\mu_{t+1} + \sup_{\pi_t \in \Pi^M(\mu_t, \mu_{t+1})} \int \widetilde c_t(x, y)\,d\pi_t(x, y)$$ where $V_t$ is the continuation value at time $t$ and $\widetilde c_t$ is as above. This DP structure generalizes the frictionless case and serves as a key principle for both theoretical and numerical analysis.

Stability results show that MOT solutions (plans and transport graphs) are continuous and robust under weak or Wasserstein convergence of marginals, and under perturbation of the cost (including vanishing friction). In particular, as friction vanishes, frictional MOT optimizers converge to the left-curtain solution of the frictionless problem (Rai, 9 Oct 2025).

5. Explicit Formulas and Comparative Statics: The Linear-Quadratic Case

In the linear--quadratic case ($f(v) = \alpha|v| + \beta v^2$), explicit closed-form expressions are available for off-band displacements: $$y - x = \frac{h(x) - \alpha\, \mathrm{sgn}(h(x))}{2 \beta}, \quad \text{for } |h(x)| > \alpha.$$ The width of the no-trade band increases monotonically with $\alpha$ and decreases with $\beta$. The conditional turnover and expected cost can be computed explicitly in terms of the two-atom support: $$\mathbb{E}[\,|Y - X| \mid X = x] = 2\,\theta(x)\,(1-\theta(x))\,[T_u(x)-T_d(x)].$$ These comparative statics enable sensitivity analysis of execution strategies to liquidity parameters and hold critical implications for implementation in robust hedging contexts.

6. Applications: Superhedging and Path-Dependent Claims

MOT under friction provides robust, model-independent price and superhedging bounds across a spectrum of path-dependent options:

• Lookback options: With state $M_t = \max_{s \leq t} S_s$, trade bands widen near the kink $y = m$ in the continuation value.
• Barrier options: Trade bands reflect regions where optimality ignores barrier crossings, as identified by the peculiar structure of kinks in continuation values.
• Asian options: The smoothness of the state $A_t = \sum_{s \leq t} S_s$ leads to unique graphs and thin bands. The dual variables $(\varphi_t, \psi_t, h_t)$ define semi-static hedging strategies (in vanilla options and dynamic trading), directly implementing the logic of robust superhedging frameworks (Rai, 9 Oct 2025).

7. Impact, Theory Integration, and Broader Developments

The frictional MOT theory unifies the geometry of left-monotone (left-curtain) couplings from the frictionless literature with robust superhedging duality developed in previous MOT works (e.g., Beiglböck–Henry-Labordère–Penkner, Dolinsky–Soner) and enriches it to accommodate liquidity and frictions (Rai, 9 Oct 2025). The monotonicity principle and no-crossing geometry extend, via the analysis of the adjusted one-step cost, to the frictional, multi-dimensional, and multi-marginal settings. Stability results support the applicability in computational and practical regimes, while explicit formulas facilitate the design and evaluation of trading strategies, robust price bounds, and comparative statics under realistic market imperfections.

These mathematical foundations link optimal transport, robust finance, dynamic programming, and the microstructure of financial markets within a common mathematical framework.