{
  "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 $f/(g\\circ T)$, 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).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27785",
  "author_total": 3,
  "authors": [
    {
      "name": "Ryan M. Engel"
    },
    {
      "name": "Kibaek Lee"
    },
    {
      "name": "Namid Stillman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5npfgrezb47tj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27785",
  "fulltext_url": null,
  "id": "5npfgrezb47tj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27785",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CE"
    ],
    [
      "arxiv",
      "q-fin.CP"
    ],
    [
      "arxiv",
      "q-fin.RM"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7214195734",
  "pdf_url": "https://arxiv.org/pdf/2609.27785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5npfgrezb47tj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:06Z",
      "native_id": "2609.27785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport",
  "title_text": "Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:06Z",
  "url": "https://research.draconitic.com/paper/5npfgrezb47tj",
  "venue": "arXiv",
  "versions": []
}