Paper: arXiv 2604.10492

Authors: Takanori Adachi

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly novel, sophisticated mathematical framework linking category theory, algebraic topology, and homological obstructions to financial arbitrage. However, it is entirely theoretical with zero empirical validation, backtesting, or real-world data implementation.

Research Flowchart

  flowchart TD
    A["Research Goal: Formulate Global Arbitrage via Cohomological Structures & Homology"] --> B["Input: Filtration as Contravariant Functor F: T^op → Prob"]
    B --> C["Compute Multiplicative Distortion dF(i) via Conditional Expectations"]
    C --> D["Define Holonomy Along Loops in Market Topos T"]
    D --> E{"Is Holonomy Non-Trivial?"}
    E -- Yes --> F["Construct Predictable Self-Financing Trading Strategy"]
    E -- No --> G["Market Free of Global Loop Inconsistencies"]
    F --> H["Outcome: Aharonov-Bohm AB Arbitrage Realized from Global Invariants"]