超越Lipschitz连续性的金融尾部风险：基于半离散最优传输
Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport
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中文总结 AI 辅助
针对金融重尾分布，提出半离散最优传输方法，放宽映射正则性以精确估计尾部风险，实验显示优于神经生成器，在CVaR优化下获得最佳风险调整收益。
中文摘要 AI 辅助
金融收益率具有重尾特征，准确的尾部风险估计是投资组合风险管理的核心。现代神经生成器通过学习到的映射将简单基分布推向前进，为保持训练稳定性，该映射由Lipschitz分量构成。这是一个约束条件：高斯分布的Lipschitz映射是亚高斯的，因此更重尾的目标在任意有限Lipschitz常数下都无法精确匹配。Monge–Ampère方程将Brenier映射的局部扭曲与密度比$f/(g\circ T)$联系起来，因此目标密度中更深的谷值需要更高增益的映射，并产生更高方差的估计器。该论证仅需有界扭曲，因此同样适用于标准化流、流匹配、生成对抗网络和扩散采样器。\n 半离散最优传输（SDOT）放宽了映射的正则性而非源的尾部类别。其功率图赋予每个训练观测一个恰好承载源测度$1/N$的单元，尾部观测通过跨越单元边界而非拉伸来达到。我们的主要实验在标定的Merton跳跃扩散上扫描严重程度，峰度范围从94到1,679。SDOT将尾部比率保持在$0.85$–$0.94$，跨种子标准差低于$0.025$，而每个学习生成器要么压缩尾部，要么以随之增长的方差膨胀尾部。进一步实验将结果推广至真实的标准普尔500指数收益率和21年回测，在CVaR优化下，SDOT给出了最佳风险调整的市场中性策略（夏普比率$0.70$，最大回撤$-2.60\\%$，而次优生成器为$0.40$）。
英文摘要
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).
发表机构
- Stony Brook University（石溪大学）
- Simudyne
机构由 AI 辅助整理，请以论文原文为准。
