Convex Order and Arbitrage

2025-10-01

AI Summary

This paper investigates the intrinsic connection between convex order relations among probability measures and the no-arbitrage condition in financial markets. Methodologically, it employs optimal transport theory to characterize convex order via the Wasserstein distance and constructs a unique optimal coupling ρ and its associated convex potential function using Brenier’s theorem. Building on this, the paper establishes an arbitrage criterion linking ρ to functionally generated portfolios, thereby proposing the first model-free, convex-order-driven framework for constructing arbitrage strategies. A novel numerical algorithm—based on ρ-optimization—is designed to efficiently compute such couplings. Empirical validation via stochastic portfolio experiments confirms that whenever marginal measures violate convex order, a model-free arbitrage strategy with strictly positive payoff and zero initial cost can always be constructed. The work extends both the theoretical foundations and practical applicability of convex order in mathematical finance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Security and Privacy: Large-scale security measurements

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. $μpreceq_c ν$) 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.

Problem

• Characterizing convex order via optimal transport inequalities
• Developing algorithms to find optimal transport measures
• Constructing arbitrage strategies using convex order relationships

Innovation

• Develops new algorithms for optimal transport measures
• Constructs model-independent arbitrage strategy via Brenier's theorem
• Links convex order to arbitrage in portfolio theory