Against a Universal Trading Strategy: No-Arbitrage, No-Free-Lunch, and Adversarial Cantor Diagonalization

TL;DR

This paper proves fundamental limits on universal trading strategies across measure-theoretic, combinatorial, and computational models, showing such strategies cannot guarantee profits in general market conditions.

Contribution

It introduces a unified framework demonstrating the impossibility of universal profit-generating trading strategies using measure theory, combinatorics, and Turing diagonalization.

Findings

Universal trading strategies are strictly limited by no-arbitrage principles.

Outperformance requires exploiting non-uniform market structures.

Adversarial environments can defeat any computable trading algorithm.

Abstract

We investigate the impossibility of universally winning trading strategies -- those generating strict profit across all market trajectories -- through three distinct mathematical paradigms. Fundamentally, under standard admissibility constraints, the existence of such a strategy is a strict subset of strong arbitrage, which is mathematically precluded in competitive markets admitting an equivalent martingale measure. Beyond this rigorous measure-theoretic foundation, we explore analogous limitations in two alternative modeling regimes. Combinatorially, the No-Free-Lunch theorem demonstrates that outperformance requires exploitation of non-uniform market structure, as uniform averaging precludes universal dominance. Computationally, a Turing diagonalization argument constructs an adversarial environment that defeats any computable trading algorithm, shifting the impossibility from exogenous price paths to adaptive adversaries. These mathematical limits are framed by a time-reversal heuristic that establishes a formal analogy between financial martingale measures and thermodynamic detailed balance, resolving the Maxwell's Demon analogy for markets without relying on physically irrelevant Landauer erasure costs. Using the Wheel Options Strategy as a case study, we demonstrate that strategies succeeding ``for all practical purposes'' (FAPP) inherently depend on transient regime assumptions, meaning their automated execution systematically amplifies tail risks.