Paper: arXiv 2309.04947

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

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous mathematical proofs establishing a single-factor structural reduction in two-asset Vectorial Martingale Optimal Transport (VMOT). While the theoretical novelty is exceptional and backed by numerical Sinkhorn experiments, empirical validation on real market option data remains limited.

Research Flowchart

  flowchart TD
    A["Research Goal: Robust Option Pricing via<br>Vectorial Martingale Optimal Transport (VMOT)"] --> B["Inputs: Assets (d), N-periods, &<br>Sub-/Supermodular Payoffs"]
    B --> C["Geometric Analysis of<br>VMOT Extremal Models"]
    C --> D{"Asset Dimension (d)?"}
    D -- "d = 2" --> E["Dimension Reduction:<br>1-Factor Structure in Period 1"]
    D -- "d ≥ 3" --> F["Failure of Reduction<br>(Counterexamples)"]
    E --> G["Reduced-Dimension<br>Sinkhorn Algorithm"]
    G --> H["Outcomes: ~99% Reduced Compute Time<br>& Higher Pricing Accuracy"]