Financial Tail Risk Beyond Lipschitz Continuity via Semi-Discrete Optimal Transport

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--Amp`ere equation ties the Brenier map's local distortion to the density ratio f/(gcirc 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).