Fin Return Paradox

Every formula in quantitative finance — CAPM, Markowitz, VaR, Sharpe ratio, GARCH — takes returns as input. Yet the standard definitions of return fail when prices cross zero: log-returns are undefined, and simple returns produce sign errors. This is not hypothetical: WTI crude settled at \(-\$37.63\) on April 20, 2020, and European electricity markets routinely produce negative prices. We ask: what properties must a universal return function satisfy? We prove that six axioms — five structural (additivity, smoothness, monotonicity, log-asymptotic behavior, antisymmetry) and one for transparency (algebraic closure, ensuring both directions of the price-return mapping are elementary) — determine the return function uniquely up to a single scale parameter. The solution is the arcsinh-return: \( R(S, S') = \operatorname{arcsinh}(S') - \operatorname{arcsinh}(S) \). This is the only return function that (i) works for all prices \( S \in \mathbb{R} \), (ii) recovers log-returns for large positive prices, (iii) is dimensionless, and (iv) has a closed-form inverse (\(\sinh\)). The arcsinh-return implies a volatility measure that remains well-defined through zero-crossing events, unlike log-return-based Black-volatility workflows, which require dropping observations or switching models. We derive implications for risk measurement (arcsinh-VaR, arcsinh-GARCH), regulatory compliance (FRTB), and performance attribution. The core axioms, return properties, and key characterization lemmas are formally verified in Lean 4; one elementary rational-function step in the uniqueness argument remains handwritten.