用稳定分布替代高斯分布，提升金融风险建模的准确性
Lévy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management
- 用方差伽马和正态逆高斯分布代替高斯分布，捕捉金融数据的重尾特征
- 在标普500日收益率上，负对数似然降低69%，95%风险价值计算精确匹配
- 适合关注极端风险、需精准估计尾部损失的量化分析师和风控系统
我们提出Lévy-Flows，一类将标准高斯基分布替换为基于莱维过程的分布（如方差伽马VG和正态逆高斯NIG）的归一化流模型。这些分布能自然刻画重尾特性，同时保持精确似然评估和高效重参数化采样。理论上，我们证明了对于规则变化基分布，其尾指数在渐近线性流变换下得以保留；且身份尾部神经样条流架构在变换区域外可完全保持基分布的尾部形状。实证上，在标普500日收益率及其他资产上评估，显示密度估计与风险校准均有显著提升：基于VG的流模型使测试负对数似然相比高斯流降低69%，并实现精确的95% VaR校准；基于NIG的流模型则给出最准确的期望损失估计。结果表明，将莱维过程结构融入归一化流，能显著提升重尾数据建模能力，适用于金融风险管理。
原文摘要 · Abstract (English)
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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