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

Joshua Zoen-Git Hiew, Tongseok Lim, Brendan Pass, Marcelo Cruz de Souza

TL;DR

The paper analyzes robust option pricing via Vectorial Martingale Optimal Transport (VMOT) in multi-period, two-asset settings, showing that for $d=2$ and supermodular costs, the first-period joint distribution of the assets is monotone, enabling a dimension reduction that keeps the martingale structure intact. It develops a dual formulation and proves dual attainment, then uses convex analysis to establish the main monotonicity result, plus a perturbation argument to handle strict supermodularity; counterexamples demonstrate sharpness in higher dimensions ($d\ge 3$). The authors introduce a reduced-dimension Sinkhorn algorithm that exploits the monotone first-period structure, achieving up to roughly $99\%$ reductions in compute time while maintaining or improving accuracy, and they present several numeric experiments across two- and three-period problems. The work clarifies when dimension reduction is possible in VMOT, contrasts VMOT with MMOT, and provides practical guidance for robust pricing with scalable computation in the two-asset regime.

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.

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

TL;DR

The paper analyzes robust option pricing via Vectorial Martingale Optimal Transport (VMOT) in multi-period, two-asset settings, showing that for and supermodular costs, the first-period joint distribution of the assets is monotone, enabling a dimension reduction that keeps the martingale structure intact. It develops a dual formulation and proves dual attainment, then uses convex analysis to establish the main monotonicity result, plus a perturbation argument to handle strict supermodularity; counterexamples demonstrate sharpness in higher dimensions (). The authors introduce a reduced-dimension Sinkhorn algorithm that exploits the monotone first-period structure, achieving up to roughly reductions in compute time while maintaining or improving accuracy, and they present several numeric experiments across two- and three-period problems. The work clarifies when dimension reduction is possible in VMOT, contrasts VMOT with MMOT, and provides practical guidance for robust pricing with scalable computation in the two-asset regime.

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 -period market models and demonstrate that, when the number of underlying assets is 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 . 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.
Paper Structure (23 sections, 5 theorems, 62 equations, 4 figures, 3 tables)

This paper contains 23 sections, 5 theorems, 62 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main Theorem and Conjecture
  3. Main Theorem
  4. The Conjecture
  5. Motivation and Heuristic
  6. Examples of Supermodular Costs
  7. Dual Problem, Dual Attainment and Proof of the Main Theorem
  8. Dual Problem
  9. Dual Attainment
  10. Proof of the Main Theorem
  11. Counterexamples and Related Examples
  12. Counterexamples to the Strict Supermodularity of the Cost
  13. Counterexamples to the conjecture
  14. Example for Tighter Bounds for Single-Period Costs in VMOT
  15. Example: Relaxing the Martingale Constraints: Marginal Martingale Transport
  16. ...and 8 more sections

Key Result

Theorem 2.3

\newlabelmain0 Let $d = 2$ and $N \geq 2$. Consider the cost function $c(x_1, \dots, x_N) = \sum_{t=1}^N c_t(x_{t,1}, x_{t,2})$, where each $c_t$ is a supermodular function. Assume the following conditions: Then:

Figures (4)

  • Figure 1: Conjecture \ref{['conjecture1']} asserts that the martingale transport in Figure \ref{['goodmart']} can be superior to the one in Figure \ref{['badmart']} for the problem \ref{['VMOT']} due to the supermodularity of $c_1, c_2$. We overlap ${\rm Law}(X)$ with ${\rm Law}(Y)$ in Figure \ref{['goodmart']} to emphasize that they must be in convex order.
  • Figure 1: $\beta_0$ values on the vertices.
  • Figure 1: Heat maps of the induced couplings between different marginals obtained from the reduced (top row) and complete (bottom row) Sinkhorn algorithms. Lighter colors indicate higher probability density.
  • Figure 2: Heat maps of the induced couplings between different marginals obtained from the Sinkhorn algorithm for the three‐period case. Lighter colors indicate higher probability density. The figure compares the complete and reduced formulations discussed in the text.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Conjecture 2.6
  • Theorem 2.7
  • Example 2.8
  • Remark 2.9
  • Theorem 3.1: Lim23
  • ...and 8 more