Aharanov-Bohm Type Arbitrage and Homological Obstructions in Financial Markets

Abstract: We introduce a new notion of arbitrage based on global loop effects in filtered market systems. 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 provides a conceptual link between cohomological structures and economically realizable arbitrage opportunities.

• The paper introduces a novel arbitrage framework using nontrivial loop holonomy in market filtrations detected via multiplicative cocycles.
• It employs a categorical model to capture global market information, bridging conditional expectations with cohomological obstructions.
• The framework translates global holonomy into executable, self-financing trading strategies, expanding the classical no-arbitrage theory.

Aharonov–Bohm Type Arbitrage and Homological Obstructions in Financial Markets

Overview and Motivation

This work introduces a formalism for arbitrage in financial markets predicated on global, rather than local, features of filtered probability spaces. Inspired by the Aharonov–Bohm (AB) effect in quantum physics—where a particle’s phase shift accumulates nontrivially along a closed loop in the presence of vanishing local fields—the analysis shifts focus from classical local conditions (such as those enforced by equivalent martingale measures) to global loop-based inconsistencies in the probabilistic evolution of market systems.

The filtration underpinning the market information is modeled categorically as a contravariant functor $F : \mathcal{T}^{op} \to \mathrm{Prob}$, associating to arrows in an index category $\mathcal{T}$ transitions between probability spaces. The paper’s primary technical contributions center on defining and studying a multiplicative cocycle—termed distortion—arising from the failure of conditional expectations, induced by market transitions, to preserve constants. Nontrivial loop holonomy of this cocycle is posited as a novel and fundamentally global arbitrage mechanism.

Categorical Framework and Distortion

The transition structure of financial information is captured by a small category $\mathcal{T}$; the market filtration is specified as a contravariant functor $F : \mathcal{T}^{op} \to \mathrm{Prob}$, with $\mathrm{Prob}$ the category of probability spaces and null-set-preserving measurable maps. Conditional expectations, formalized via a functor $\mathcal{E} : \mathrm{Prob} \to \mathrm{Ban}$ (the latter the category of Banach spaces), are composed with $F$ to yield $$\mathcal{E} \circ F : \mathcal{T}^{op} \to \mathrm{Ban}$$.

A key insight is that, generically, the iterated conditional expectation $(\mathcal{E} \circ F)(i)$ associated with an arrow $i : s \to t$ need not preserve constant functions. The deviation is captured by the multiplicative distortion:

$\mathcal{T}$0

which measures the breakdown of measure-preservation in these transitions. The distortion $\mathcal{T}$1 satisfies a multiplicative 1-cocycle property under composition of arrows—a structure highly reminiscent of holonomy in gauge theories or vector potentials, providing a categorical-geometric flavor to the information dynamics of the market.

Loop Holonomy and Definition of AB Arbitrage

For a loop $\mathcal{T}$2, with each $\mathcal{T}$3 and $\mathcal{T}$4, the holonomy

$\mathcal{T}$5

is defined by recursively transporting and composing the sequence of distortions along $\mathcal{T}$6, referencing all random variables back to the base $\mathcal{T}$7. This cumulative effect, analogous to the path-ordered exponential in physics, encodes the system’s total deviation from local consistency upon traversing the loop.

AB arbitrage is defined by the existence of a loop $\mathcal{T}$8 such that

$\mathcal{T}$9

where $\mathcal{T}$0 is the measure associated with $\mathcal{T}$1. This condition asserts the possibility of positive expected gain for a self-financing round-trip strategy, even when all one-step transitions are locally consistent.

The construction is demonstrated via an explicit categorical/probabilistic example, showing how information loss and reinjection, when not globally measure-preserving, generate nontrivial loop holonomy and thus admit arbitrage in the AB sense.

From Holonomy to Executable Trading Strategies

A critical technical result is the translation of nontrivial holonomy into economically realizable, predictable, self-financing trading strategies. The design is straightforward and operational: at initial time $\mathcal{T}$2, based entirely on $\mathcal{T}$3-measurable quantities, the strategy consists of (i) executing $\mathcal{T}$4 if $\mathcal{T}$5, (ii) executing the reverse loop $\mathcal{T}$6 if $\mathcal{T}$7, and (iii) abstaining otherwise.

Formally, the strategy is encoded by a position function $\mathcal{T}$8, which is $\mathcal{T}$9-measurable and supports predictable execution. The resulting wealth at terminal time is explicitly determined by the loop holonomy, and the construction guarantees nonnegative payoff with strictly positive probability of gain when the holonomy condition is met.

Admissibility: Microstructure Constraints and Practicality

Not all categorical loops correspond to viable financial strategies; the notion of admissibility is introduced to formalize when a loop-based holonomy can be exploited:

• The holonomy must be observable/predictable at the base time.
• Each morphism in the loop must correspond to an implementable market operation.
• The sequence must be composable, enabling sequential execution.
• The entire strategy must be self-financing, i.e., require no additional capital after initialization.
• Reversal of the loop must also be executable for strategies exploiting negative holonomy.

Admissibility serves to separate the geometric, cohomological properties of filtrations from the market microstructure, and is essential for the translation from global structural obstruction to implementable arbitrage.

Theoretical and Practical Implications

The approach unearths a fundamentally new class of arbitrage opportunities—those invisible to classical local-martingale-based conditions—by leveraging higher-order, cohomological structures in market filtrations. The presence of nontrivial multiplicative cocycles (holonomy) is shown to both detect and construct explicit arbitrage strategies in complex market models.

Future directions suggested by this framework include:

• Development of a full cohomological theory of filtered market systems.
• Precise formulation and characterization of global no-arbitrage conditions as triviality of certain categorical invariants.
• Integration of market frictions and execution constraints into the admissibility criterion.
• Revisiting the Fundamental Theorem of Asset Pricing under this extended, homological viewpoint and understanding when classical and AB-type arbitrage conditions are equivalent or distinct.

The categorical and cohomological methodology is robust to the choice of time indexation, extending to non-discrete or context-dependent filtrations as well [see also "A geometric model of synthetic filtrations via context-dependent time", (Adachi, 16 Sep 2025)]. This aligns with the general trend of adopting advanced algebraic and categorical tools in mathematical finance for structurally complex or path-dependent markets.

Conclusion

This work formalizes a novel, globally-oriented concept of arbitrage—Aharonov–Bohm type—arising from nontrivial holonomy in categorical filtrations of market information. The construction provides both theoretical insights, illuminating new homological obstructions to arbitrage-free markets, and applicable tools for building self-financing arbitrage strategies based on global market properties. The framework bridges categorical probability, cohomology, and finance, and delineates a new research agenda in the structure and detection of arbitrage in complex systems.