Levy-stable scaling of risk and performance functionals

TL;DR

This paper introduces a model capturing Levy-stable scaling in asset returns over a specific window, providing horizon-adjusted formulas for risk and performance metrics that are applicable across different investment horizons.

Contribution

It develops a nonparametric, horizon-corrected framework for risk and performance functionals based on Levy-stable scaling, with closed-form bias adjustments.

Findings

Formulas for VaR, ES, Sharpe, and other ratios are derived and validated.

The model accurately captures Levy-stable behavior within the identified window.

Results are consistent across different horizons on the Levy window.

Abstract

We develop a finite-horizon model in which liquid-asset returns exhibit Levy-stable scaling on a data-driven window [tau_UV, tau_IR] and aggregate into a finite-variance regime outside. The window and the tail index alpha are identified from the log-log slope of the central body and a two-segment fit of scale versus horizon. With an anchor horizon tau_0, we derive horizon-correct formulas for Value-at-Risk, Expected Shortfall, Sharpe and Information ratios, Kelly under a Value-at-Risk constraint, and one-step drawdown, where each admits a closed-form Gaussian-bias term driven by the exponent gap (1/alpha - 1/2). The implementation is nonparametric up to alpha and fixed tail quantiles. The formulas are reproducible across horizons on the Levy window.