Aharanov-Bohm Type Arbitrage and Homological Obstructions in Financial Markets

TL;DR

This paper introduces a novel arbitrage concept based on global loop effects and homological obstructions, extending classical theory through topological and cohomological insights.

Contribution

It develops a framework linking holonomy and cohomology to arbitrage, revealing global invariants and loop effects as sources of arbitrage in financial markets.

Findings

Holonomy along loops measures global market inconsistencies.

Non-trivial holonomy can be transformed into predictable trading strategies.

Global invariants reveal arbitrage opportunities invisible to local analysis.

Abstract

We introduce a new perspective on arbitrage based on global loop effects in filtered market systems, providing a conceptual extension of classical arbitrage theory beyond local consistency conditions. Given a filtration modeled as a contravariant functor $F : \mathcal{T}^{op} \to \mathrm{Prob}$, we consider the associated conditional expectation functor $\mathcal{E} \circ F$ and show that it induces a canonical multiplicative distortion $dF(i) := (\mathcal{E} \circ F)(i)(1)$, which measures the failure of constant functions to be preserved under non-measure-preserving transitions. We define the holonomy of $dF$ along loops in $\mathcal{T}$ and interpret non-trivial holonomy as a global inconsistency that is invisible at the level of individual transitions. This leads to a notion of Aharonov--Bohm (AB) arbitrage, in which arbitrage arises from loop effects rather than local price discrepancies. We further show that, under suitable admissibility conditions, non-trivial holonomy can be converted into a predictable self-financing trading strategy. This establishes a connection between cohomological structures and economically realizable arbitrage, highlighting the role of global invariants in the structure of financial markets.