Dimension Reduction in Martingale Optimal Transport: Geometry and Robust Option Pricing

TL;DR

This paper explores dimension reduction in vectorial martingale optimal transport for robust option pricing, revealing a single-factor structure in two-asset models that significantly simplifies computations and enhances efficiency.

Contribution

It identifies a structural simplification for two-asset models under sub- or supermodular payoffs, enabling a reduced-dimension approach and a more efficient Sinkhorn algorithm.

Findings

Dimension reduction to a single-factor model for two assets with sub- or supermodular payoffs.

The reduction does not extend to three or more assets, with counterexamples provided.

Numerical experiments show 99% faster computation and improved accuracy using the reduced model.

Abstract

This paper addresses the problem of robust option pricing within the framework of Vectorial Martingale Optimal Transport (VMOT). We investigate the geometry of VMOT solutions for $N$-period market models and demonstrate that, when the number of underlying assets is $d=2$ and the payoff is sub- or supermodular, the extremal model reduces to a single-factor structure in the first period. This structural result allows for a significant dimension reduction, transforming the problem into a more tractable format. We prove that this reduction is specific to the two-asset case and provide counterexamples showing it generally fails for $d \geq 3$. Finally, we exploit this monotonicity to develop a reduced-dimension Sinkhorn algorithm. Numerical experiments demonstrate that this structure-preserving approach reduces computational time by approximately 99\% compared to standard methods while improving accuracy.