Caratheodory, Finite Resources and the Geometry of Arbitrage

TL;DR

This work reframes no-arbitrage in finance as a geometric constraint rooted in Carathéodory’s adiabatic inaccessibility and argues that finite-resource describability forces markets to lie in the exponential family. By invoking the Pitman–Koopman–Darmois theorem, the authors show that a fixed-dimensional sufficient statistic is necessary for global coherence, yielding a globally convex potential and a well-defined Fisher metric that suppresses long-scale arbitrage loops. The analysis uses Boyling’s counterexample to illustrate the breakdown of local consistency without the exponential-family structure, and connects equilibrium geometry to dynamics via Onsager reciprocity and gradient-flow arguments. The resulting framework unifies thermodynamic and financial structure under a shared information-theoretic geometry, while highlighting regimes (small systems, critical points, regime shifts) where the model’s assumptions may fail and require alternative descriptions.

Abstract

Caratheodory's axiom of adiabatic inaccessibility states that, in any neighborhood of a thermodynamic state, certain states remain unreachable via adiabatic processes. Non-arbitrage mirrors this topological restriction in finance. Preserving this constraint in resource-limited systems identifies the exponential family not as a modeling convenience but as the requisite geometric structure unifying both domains.