Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport
Abstract
Financial returns are heavy-tailed, and accurate tail risk estimation is central to portfolio risk management. Modern neural generators sample by pushing a simple base distribution through a learned map, and for training stability that map is built from Lipschitz components. This is the binding constraint: a Lipschitz map of a Gaussian is sub-Gaussian, so heavier-tailed targets admit no exact match at any finite Lipschitz constant. The Monge--Ampère equation ties the Brenier map's local distortion to the density ratio f/(g∘ T), so a deeper trough in the target density requires a higher-gain map and yields a higher-variance estimator. The argument needs only bounded distortion, so it covers normalizing flows, flow matching, GANs, and diffusion samplers alike. Semi-Discrete Optimal Transport (SDOT) relaxes the map's regularity rather than the source's tail class. Its power diagram gives every training observation a cell holding exactly 1/N of the source measure, and tail observations are reached by crossing a cell boundary rather than by stretching. Our primary experiment sweeps severity over a calibrated Merton jump-diffusion spanning kurtosis 94 to 1,679. SDOT holds tail ratios at $0.85−−0.94$ with cross-seed standard deviations below $0.025$, while every learned generator either compresses the tails or inflates them with a variance that grows alongside. Further experiments carry the result to real S\&P~500 returns and to a 21-year backtest, where SDOT gives the best risk-adjusted market-neutral strategy under CVaR optimization (Sharpe $0.70$, max drawdown −2.60%, against $0.40$ for the next-best generator).
Related
- Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
- Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems
- GPU-Accelerated Bregman Douglas-Rachford Splitting for Discrete Optimal Transport
- A Network Simplex Framework for Discrete Unbalanced Optimal Transport
- Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime
BibTeX
@misc{engel2026financial,
title = {{Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport}},
author = {Ryan M. Engel and Kibaek Lee and Namid Stillman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27785},
eprint = {2609.27785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27785},
}
RIS
TY - UNPB
TI - Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport
AU - Ryan M. Engel
AU - Kibaek Lee
AU - Namid Stillman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27785
UR - https://arxiv.org/abs/2609.27785
AB - Financial returns are heavy-tailed, and accurate tail risk estimation is central to portfolio risk management. Modern neural generators sample by pushing a simple base distribution through a learned map, and for training stability that map is built from Lipschitz components. This is the binding constraint: a Lipschitz map of a Gaussian is sub-Gaussian, so heavier-tailed targets admit no exact match at any finite Lipschitz constant. The Monge--Ampère equation ties the Brenier map's local distortion to the density ratio f/(g∘ T), so a deeper trough in the target density requires a higher-gain map and yields a higher-variance estimator. The argument needs only bounded distortion, so it covers normalizing flows, flow matching, GANs, and diffusion samplers alike. Semi-Discrete Optimal Transport (SDOT) relaxes the map's regularity rather than the source's tail class. Its power diagram gives every training observation a cell holding exactly 1/N of the source measure, and tail observations are reached by crossing a cell boundary rather than by stretching. Our primary experiment sweeps severity over a calibrated Merton jump-diffusion spanning kurtosis 94 to 1,679. SDOT holds tail ratios at $0.85−−0.94$ with cross-seed standard deviations below $0.025$, while every learned generator either compresses the tails or inflates them with a variance that grows alongside. Further experiments carry the result to real S\&P~500 returns and to a 21-year backtest, where SDOT gives the best risk-adjusted market-neutral strategy under CVaR optimization (Sharpe $0.70$, max drawdown −2.60%, against $0.40$ for the next-best generator).
ER -