Dual Attainment in Multi-Period Multi-Asset Martingale Optimal Transport and Its Computation

TL;DR

This work advances the mathematical theory and practical computation of vectorial martingale optimal transport by proving dual attainment for multi-asset, multi-period MOT under mild regularity and irreducibility, and by implementing a scalable PDLP-based framework to solve large discrete VMOT problems. Theoretical results guarantee the existence of dual optimizers and enable pathwise replication or sub-/super-hedging of complex, path-dependent payoffs in high dimensions. Numerically, the authors demonstrate near-optimal solutions and tight constraint satisfaction for a worst-of autocallable option using GPU-accelerated PDLP, illustrating the practical relevance for robust pricing and hedging in model-uncertain markets. Collectively, the paper bridges rigorous MOT duality with actionable computational methods, informing robust financial engineering in high-dimensional settings.

Abstract

We establish dual attainment for the multimarginal, multi-asset martingale optimal transport (MOT) problem, a fundamental question in the mathematical theory of model-independent pricing and hedging in quantitative finance. Our main result proves the existence of dual optimizers under mild regularity and irreducibility conditions, extending previous duality and attainment results from the classical and two-marginal settings to arbitrary numbers of assets and time periods. This theoretical advance provides a rigorous foundation for robust pricing and hedging of complex, path-dependent financial derivatives. To support our analysis, we present numerical experiments that demonstrate the practical solvability of large-scale discrete MOT problems using the state-of-the-art primal-dual linear programming (PDLP) algorithm. In particular, we solve multi-dimensional (or vectorial) MOT instances arising from the robust pricing of worst-of autocallable options, confirming the accuracy and feasibility of our theoretical results. Our work advances the mathematical understanding of MOT and highlights its relevance for robust financial engineering in high-dimensional and model-uncertain environments.

Figures (4)

• Figure 1: Discrete marginal distributions used as input to the $2$-dimensional, $3$-time-step MOT problem. Asset prices are scaled by their respective spot prices at $t_0$.
• Figure 2: Absolute infeasibility with respect to the marginal constraints in the primal solution of the $2$-dimensional, $3$-time-step MOT problem. Results are shown for both maximization and minimization directions, computed using the PDLP solver in NVIDIA cuOpt with an optimality tolerance of $10^{-12}$. The low levels of infeasibility confirm the solver's ability to enforce marginal constraints with high precision.
• Figure 3: Absolute infeasibility with respect to the martingale constraints in the primal solution of the $2$-dimensional, $3$-time-step MOT problem. The PDLP solver in cuOpt maintains infeasibility below the grid discretization error for both optimization directions, demonstrating robust enforcement of path-dependent martingale constraints.
• Figure 4: Absolute infeasibility with respect to the dual constraints in the primal solution of the $2$-dimensional, $3$-time-step MOT problem. The results highlight the tightness of the dual bounds achieved by the PDLP solver in cuOpt, with infeasibility consistently below the prescribed tolerance.

Theorems & Definitions (8)

• Definition 2.1: Vectorial Martingale Transports
• Theorem 3.1
• Proposition 3.2
• Lemma 3.3
• Definition 5.1: Multi-Asset Autocall Payoff
• proof : Proof of Lemma \ref{['L1bound*']}
• proof : Proof of Proposition \ref{['ptwiseconverge']}
• proof : Proof of Theorem \ref{['main']}