L\'evy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management

TL;DR

This paper introduces LévY-Flow models that incorporate heavy-tailed LévY process distributions into normalizing flows, improving financial risk modeling by better capturing tail behavior.

Contribution

It proposes a novel class of normalizing flows using LévY process-based distributions, with theoretical guarantees and empirical validation on financial data.

Findings

VG-based flows reduce negative log-likelihood by 69% compared to Gaussian flows.

LévY-Flow models achieve exact 95% VaR calibration.

NIG-based flows provide the most accurate Expected Shortfall estimates.

Abstract

We introduce L\'evy-Flows, a class of normalizing flow models that replace the standard Gaussian base distribution with L\'evy 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 L\'evy process structure into normalizing flows yields significant gains in modeling heavy-tailed data, with applications to financial risk management.