Lévy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management
This addresses financial risk management by improving density estimation and risk calibration for heavy-tailed data, though it is incremental as it adapts existing normalizing flow methods with new base distributions.
The paper tackled the problem of modeling heavy-tailed financial data by introducing Lévy-Flow models, which replace Gaussian base distributions with Lévy process-based ones, resulting in a 69% reduction in test negative log-likelihood and exact 95% VaR calibration compared to Gaussian flows.
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.