Convex Order and Arbitrage

Abstract

Wiesel and Zhang [2023] established that two probability measures $\mu,\nu$ on $\mathbb{R}^d$ with finite second moments are in convex order (i.e. $\mu \preceq_c \nu$) if and only if $W_2(\nu,\rho)^2-W_2(\mu,\rho)^2 \leq \int |y|^2\nu(dy) - \int |x|^2\mu(dx).$ Let us call a measure $\rho$ maximizing $W_2(\nu,\rho)^2-W_2(\mu,\rho)^2$ the optimal $\rho$. This paper summarizes key findings by Wiesel and Zhang, develops new algorithms enhancing the search of optimal $\rho$, 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 $\rho$ 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.