Multi-Period Martingale Optimal Transport: Classical Theory, Neural Acceleration, and Financial Applications

TL;DR

This work develops a comprehensive framework for Multi-Period Martingale Optimal Transport (MMOT) that combines explicit quantitative convergence rates with algorithmic guarantees and market realism. It establishes strong duality and explicit convergence constants, provides a linear-rate Martingale-Sinkhorn solver with a sharp continuous-time bound, and introduces a transformer-based neural solver with physics-informed training that delivers up to a 1,597× online speedup while maintaining martingale constraints to $10^{-6}$. The framework supports transaction-cost-aware pricing and hedging, robustness to marginal perturbations, and calibration uncertainty, with thorough validation on 12{,}000 synthetic instances and 120 real-market scenarios. A hybrid neural-plus-Newton solver achieves production-grade accuracy and speed, enabling real-time pricing, risk dashboards, and intraday recalibration in practical financial settings. The results offer a production-ready, model-free pricing approach that reduces model risk while delivering real-time performance for complex path-dependent derivatives.

Abstract

This paper develops a computational framework for Multi-Period Martingale Optimal Transport (MMOT), addressing convergence rates, algorithmic efficiency, and financial calibration. Our contributions include: (1) Theoretical analysis: We establish discrete convergence rates of $O(\sqrt{Δt} \log(1/Δt))$ via Donsker's principle and linear algorithmic convergence of $(1-κ)^{2/3}$; (2) Algorithmic improvements: We introduce incremental updates ($O(M^2)$ complexity) and adaptive sparse grids; (3) Numerical implementation: A hybrid neural-projection solver is proposed, combining transformer-based warm-starting with Newton-Raphson projection. Once trained, the pure neural solver achieves a $1{,}597\times$ online inference speedup ($4.7$s $\to 2.9$ms) suitable for real-time applications, while the hybrid solver ensures martingale constraints to $10^{-6}$ precision. Validated on 12,000 synthetic instances (GBM, Merton, Heston) and 120 real market scenarios.

Figures (10)

• Figure 1: Computational complexity versus model risk in derivatives pricing. MMOT (green shaded) offers model-free pricing with moderate computational cost compared to Black-Scholes (high model risk) and linear programming (high complexity).
• Figure 2: Solver convergence on log-linear scale demonstrating linear convergence rate. Observed asymptotic slope $-0.065$ (blue line with markers) matches theoretical prediction $(1-\kappa^2)^{1/3} = 0.0648$ with $\kappa=0.42$ (red dashed line). Problem size: $N=10$, $M=150$, $\varepsilon=0.5$.
• Figure 3: Continuous-time convergence rate verification on log-log scale. Empirical measurements (blue circles) follow the theoretical $O(\sqrt{\Delta t})$ rate (red dashed line with slope $-0.5$). The measured slope of $-0.503$ confirms the Donsker-type bound from Theorem 3.
• Figure 4: Neural architecture: Conv1D embedding, positional encoding, 3-layer transformer (4 heads, 256 dim), dual decoder heads for potentials $u_t(x)$ and drift $h_t(x)$.
• Figure 5: Neural solver speedup factor relative to classical Sinkhorn across problem sizes. Maximum speedup of $6882\times$ observed at $(N=20, M=200)$. Performance gains vary by regime: limited by overhead for small instances and memory bandwidth for large instances.

Theorems & Definitions (25)

• Theorem 3.1: Strong Duality for Entropic MMOT
• proof
• Lemma 3.2: Constructive Feasible Point
• Remark 3.3: Comparison with Prior Work
• Theorem 4.1: Linear Convergence of Martingale-Sinkhorn
• Lemma 4.2: Martingale Projection Lipschitz
• Theorem 4.3: Improved Convergence Rate
• Corollary 4.4: Iteration Complexity
• Lemma 5.1: Donsker Rate
• Theorem 5.2: Continuous-Time Convergence