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

Rachid Drissi

cs.LGMar 31, 2026

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

Authors: Rachid Drissi

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.

Figures & tables

Figure 1: Hill estimator plot for S&P 500 returns. The estimated exponent of approximately 2.5 indicates substantially heavier tails than the Gaussian distribution. The Hill estimator assumes power-law decay; this estimate should be read as evidence that heavy-tailed bases are warranted, not as a claim about the asymptotic tail form of the fitted model.

Figure 2: Density comparison on S&P 500 returns. Lévy-Flows (VG, NIG) capture the peaked center and heavy tails better than Gaussian-based flows.

Figure 3: Tail comparison (log scale). The Lévy-Flow models maintain probability mass in the tails, while light-tailed models underestimate extreme event probabilities.

Panel of Figure 4 — Figure 4: QQ plots comparing model samples to empirical data. The Lévy-Flow (left) tracks the diagonal more closely in the tails, while the Gaussian-Flow (right) systematically underestimates extreme quantiles.

No-flow baselines (ablation)
Model	Train NLL	Test NLL	$\Delta$ vs Gaussian
VG-only	1.01 ( $<$ 0.01)	0.94 ( $<$ 0.01)	–
NIG-only	0.99 ( $<$ 0.01)	0.89 ( $<$ 0.01)	–
Flow-based models
Gaussian-Flow	1.28 ( $<$ 0.01)	1.23 ( $<$ 0.01)	0.0%
Student-t Flow	1.28 ( $<$ 0.01)	1.23 ( $<$ 0.01)	+0.4%

Table 1: Model Comparison: Negative Log-Likelihood (lower is better). Standard errors over 5 random seeds in parentheses.

Method	Viol.	Rate	Kupiec $p$	Viol.	Rate	Kupiec $p$
	95% VaR			99% VaR
Expected	25	5.0%	–	5	1.0%	–
Hist. Sim.	21	4.2%	0.40	5	1.0%	1.00
Gaussian-Flow	18	3.6%	0.13	4	0.8%	0.64
Student-t Flow	16	3.2%	0.05	3	0.6%	0.33
Lévy-Flow (VG)	25	5.0%	1.00	3	0.6%	0.33

Table 2: Backtest Results: Violation Rates and Formal Test Statistics (500 test days)

Method	95% VaR	99% VaR
Gaussian-Flow	0.16	0.02
Student-t Flow	0.53	0.01
Lévy-Flow (VG)	0.14	0.01
Lévy-Flow (NIG)	0.28	0.01

Table 3: Christoffersen Independence Test:

p

-values for VaR exceedance clustering

Confidence	Actual	Lévy-Flow (VG)	Lévy-Flow (NIG)	Student-t Flow	Gaussian
95%	-5.79%	-2.25% (0.02)	-2.76% (0.01)	-2.59% (0.01)	-2.18%
99%	-9.29%	-4.15% (0.02)	-3.50% (0.02)	-4.76% (0.09)	-3.06%
99.9%	-9.45%	-6.24% (0.07)	-5.21% (0.17)	-10.10% (0.37)	-4.04%

Table 4: VaR Comparison During 2008 Financial Crisis. Standard errors over 5 seeds in parentheses.

Confidence	Actual	Lévy-Flow (VG)	Lévy-Flow (NIG)	Student-t Flow	Gaussian
95%	-2.99%	-1.08% (0.02)	-1.66% (0.01)	-1.48% (0.01)	-1.26%
99%	-3.89%	-2.47% (0.01)	-2.47% (0.02)	-2.96% (0.04)	-1.82%
99.9%	-4.10%	-3.61% (0.03)	-3.31% (0.12)	-6.41% (0.24)	-2.45%

Table 5: VaR Comparison During 2022 Market Correction. Standard errors over 5 seeds in parentheses.

Model	ES (99%)	Underestimation
Empirical	-4.14%	–
Gaussian-Flow	-3.71%	10.4%
Student-t Flow	-5.71%	-38.2% ∗
Lévy-Flow (VG)	-4.29%	-3.8% ∗
Lévy-Flow (NIG)	-4.07%	1.6%

Table 6: Expected Shortfall Comparison at 99% Confidence

Model	S&P 500	AAPL	EEM	Gold
Gaussian-Flow	1.23 ( $<$ 0.01)	0.95 ( $<$ 0.01)	0.93 ( $<$ 0.01)	1.38 ( $<$ 0.01)
Student-t Flow	1.23 ( $<$ 0.01)	0.96 ( $<$ 0.01)	0.94 ( $<$ 0.01)	1.38 ( $<$ 0.01)
Lévy-Flow (VG)	0.38 ( $<$ 0.01)	0.10 ( $<$ 0.01)	0.08 ( $<$ 0.01)	0.52 ( $<$ 0.01)
Lévy-Flow (NIG)	0.58 ( $<$ 0.01)	0.30 ( $<$ 0.01)	0.28 ( $<$ 0.01)	0.73 ( $<$ 0.01)

Table 7: Multi-Asset Test NLL Comparison (lower is better). Standard errors over 5 seeds.

Normalizing Flows

Normalizing flows are invertible neural transformations that model complex probability distributions with tractable likelihoods and sampling.

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May 19, 2026J/K move · Enter open · S save

Tail Annealing for Heavy-Tailed Flow Matching

Jean Pachebat

1CMAP, Ecole Polytechnique, Institut Polytechnique de Paris, France.

Flow Matching Generative Modeling Flow-Based Generative Modeling Heavy-Tailed Distributions

Open paper AskPDF arXiv

Standard generative models struggle with heavy-tailed data: Lipschitz architectures cannot produce power-law tails from Gaussian noise, and interpolating between heavy-tailed data and Gaussians is ill-posed. We propose a simple fix: apply the soft-log transform

φ(x) = \mathrm{sign}(x) \cdot \log(1 + |x|)

coordinate-wise to data before training, then exponentiate samples after generation. A Hill diagnostic decides per-coordinate whether to transform, leaving light-tailed margins untouched at no added complexity. This compresses heavy tails into a range where standard flow matching succeeds, without heavy-tailed base distributions or architectural modifications. We provide theoretical intuition for why this works: the log-transform maps Pareto tails to exponentials, and the induced dynamics implement a form of tail annealing via power transformations. On a 144-configuration multivariate benchmark (3 copulas,

d

up to 100, 4 tail indices), Log-FM dominates specialized baselines on

W_1

, CVaR

_{99}

, and extreme-quantile metrics, and is the only method with zero severe divergences across 2{,}880 runs.

Abstract

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