Levy-stable scaling of risk and performance functionals

ArXiv ID: 2511.07834 “View on arXiv”

Authors: Dmitrii Vlasiuk

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.

Keywords: Lévy-stable scaling, Value-at-Risk (VaR), Expected Shortfall, Horizon correction, Nonparametric tail estimation, Liquid assets / Market Risk (General)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical machinery including Lévy-stable processes, nonparametric estimation with statistical theory, and derivations for multiple risk functionals, which justifies a high math score. It includes specific empirical steps (identifying the Lévy window via log-log slope and two-segment fit, reproducible across horizons) but lacks detailed backtesting or implementation code, placing empirical rigor above moderate.

  flowchart TD
    A["Research Goal<br>Quantify horizon-corrected risk & performance<br>for liquid assets with Lévy-stable scaling"] --> B["Data & Inputs<br>- High-frequency liquid-asset returns<br>- Horizon range [τ_UV, τ_IR"]]
    B --> C["Methodology: Lévy-Stable Scaling<br>1. Identify scaling window via log-log slope<br>2. Estimate tail index α (nonparametric)<br>3. Anchor at τ_0 for interpolation"]
    C --> D["Computational Process<br>Derive horizon-correct formulas:<br>- VaR, Expected Shortfall<br>- Sharpe & Information Ratios<br>- Kelly under VaR constraint<br>- One-step drawdown<br>All include Gaussian-bias term (1/α - 1/2)"]
    D --> E["Key Findings/Outcomes<br>- Closed-form formulas with Gaussian bias<br>- Reproducible across horizons on Lévy window<br>- Nonparametric implementation (α + tail quantiles)<br>- Unified risk & performance scaling"]