Lévy-流模型：面向金融风险管理的重尾意识归一化流
Lévy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management
AI总结：
本文提出Lévy-流模型，用Lévy过程分布替代标准高斯分布，提升重尾数据建模能力，在金融风险管理中实现更精确的风险评估和密度估计。
摘要：
我们引入Lévy-流，一种归一化流模型，将标准高斯基础分布替换为基于Lévy过程的分布，特别是方差伽玛（VG）和正态逆伽玛（NIG）分布。这些分布能自然捕捉重尾行为，同时保持精确的似然评估和高效的重参数化采样。我们建立了理论保证，证明对于正则变化的基础分布，在渐近线性流变换下尾指数得以保持，且身份尾神经流形流架构在变换区域外精确保持基础分布的尾形状。实证上，我们在S&P 500每日收益率和其他资产上评估，显示密度估计和风险校准有显著提升。基于VG的流将测试负对数似然降低69%，实现精确的95% VaR校准，而基于NIG的流提供最准确的预期短缺估计。这些结果表明，将Lévy过程结构纳入归一化流中在建模重尾数据方面有显著收益，应用于金融风险管理。
英文摘要：
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.
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