Lévy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management

2026-03-31

AI Summary

Traditional normalizing flows struggle to capture the heavy-tailed nature of financial returns, leading to biased estimates of Value-at-Risk (VaR) and Expected Shortfall (ES). This work proposes Lévy-Flow, the first framework to integrate Lévy-driven heavy-tailed distributions—specifically Variance Gamma (VG) and Normal-Inverse Gaussian (NIG)—into normalizing flows. The model explicitly captures tail behavior while preserving exact likelihood computation and enabling efficient reparameterized sampling. Theoretically, it is shown that the proposed flow maintains the tail index under asymptotically linear transformations, which motivates the design of an Identity-tail Neural Spline Flow to faithfully preserve the base distribution’s tail shape. Empirical results on S&P 500 daily returns demonstrate that the VG flow reduces test negative log-likelihood by 69% compared to Gaussian flows and achieves well-calibrated 95% VaR, while the NIG flow yields the most accurate ES estimates.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Natural Language Processing: (Large) Language Models
• Computer Vision: Large Vision Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Large language models for search
• Security and Privacy: Large-scale security measurements

Abstract

We introduce Lévy-Flows, a class of normalizing flow models that replace the standard Gaussian base distribution with Lévy process-based distributions, specifically Variance Gamma (VG) and Normal-Inverse Gaussian (NIG). These distributions naturally capture heavy-tailed behavior while preserving exact likelihood evaluation and efficient reparameterized sampling. We establish theoretical guarantees on tail behavior, showing that for regularly varying bases the tail index is preserved under asymptotically linear flow transformations, and that identity-tail Neural Spline Flow architectures preserve the base distribution's tail shape exactly outside the transformation region. Empirically, we evaluate on S&P 500 daily returns and additional assets, demonstrating substantial improvements in density estimation and risk calibration. VG-based flows reduce test negative log-likelihood by 69% relative to Gaussian flows and achieve exact 95% VaR calibration, while NIG-based flows provide the most accurate Expected Shortfall estimates. These results show that incorporating Lévy process structure into normalizing flows yields significant gains in modeling heavy-tailed data, with applications to financial risk management.

Problem

• heavy-tailed distributions
• financial risk management
• density estimation
• Value at Risk
• Expected Shortfall

Innovation

• Lévy processes
• normalizing flows
• heavy-tailed distributions
• financial risk management
• tail behavior