Convex Order and Arbitrage

Paper: arXiv 2510.01599

Authors: Erica Zhang

Abstract

Wiesel and Zhang [2023] established that two probability measures $μ,ν$ on $\mathbb{R}^d$ with finite second moments are in convex order (i.e. $μ\preceq_c ν$) if and only if $W_2(ν,ρ)^2-W_2(μ,ρ)^2 \leq \int |y|^2ν(dy) - \int |x|^2μ(dx).$ Let us call a measure $ρ$ maximizing $W_2(ν,ρ)^2-W_2(μ,ρ)^2$ the optimal $ρ$. This paper summarizes key findings by Wiesel and Zhang, develops new algorithms enhancing the search of optimal $ρ$, and builds on the paper through constructing a model-independent arbitrage strategy and developing associated numerical methods via the convex function recovered from the optimal $ρ$ through Brenier’s theorem. In addition to examining the link between convex order and arbitrage through the lens of optimal transport, the paper also gives a brief survey of functionally generated portfolio in stochastic portfolio theory and offers a conjecture of the link between convex order and arbitrage between two functionally generated portfolios.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework connecting optimal transport, convex order, and model-free arbitrage, backed by Brenier’s theorem and 2-Wasserstein metrics. While it introduces new sampling algorithms and numerical methods for 1D and 2D convex function recovery, it lacks extensive backtesting on real historical market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Characterize Convex Order μ ⪯c ν via Optimal Transport & Construct Arbitrage Strategy"] --> B["Input Data: Probability measures μ, ν on ℝᵈ with finite second moments"]
    
    subgraph Methodology & Computation
        B --> C["Maximize Wasserstein Difference:<br/>W₂(ν,ρ)² - W₂(μ,ρ)²"]
        C --> D["New Algorithms: Search for Optimal Measure ρ"]
        D --> E["Brenier's Theorem: Recover Convex Function φ from Optimal ρ"]
    End

    subgraph Key Outcomes
        E --> F["Model-Independent Arbitrage Strategy & Numerical Methods"]
        E --> G["Conjecture: Link Convex Order to Arbitrage in Functionally Generated Portfolios"]
    end

Cited References (39)

1. Inversion of Convex Ordering in the VIX Market (2019) — [link](https://ssrn.com/abstract=3504022 or http://dx.doi.org/10.2139/ssrn.3504022)
2. Variance Swap Valuation with Time-Varying Volatility and Interest Rate (2016) — link
3. Inequalities for Distributions with Given Marginals (1980) — link
4. Options on Realized Variance and Convex Orders (2018) — link
5. An optimal transport based characterization of convex order (2023) — link
6. The Existence of Probability Measures with Given Marginals (1965) — link
7. Polar factorization and monotone rearrangement of vector-valued functions (1991) — link
8. M 'e moire sur la th 'e orie des d 'e blais et des remblais (1781)

0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.