Learning to Detect Symbolic Failure: Machine Learning and the Limits of Black-Scholes

Accurate option prices do not imply accurate recovery of the latent risk-neutral density. We study this distinction with two complementary benchmarks. A controlled benchmark exposes simulator-truth densities for latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture has the lowest aggregate price, $L^1$, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators retain narrower strengths: DeepONet reduces 1% quantile and variance error by 39.0% and 34.6% relative to the mixture, and a quote transformer reduces $L^1$ by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains why these rankings can differ: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically null, and two densities separated by $L^1 = 0.061$ produce identical prices on the covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-expiry mixture and SVI fits remain much more accurate. The evidence supports target-dependent inductive bias, not a universal winner.

Modern option-learning systems operate in two coordinates: price space, where markets quote and no-arbitrage constraints are most naturally enforced, and implied volatility (IV) space, where volatility surfaces are smoothed, regularized, and evaluated. The bottleneck is interface, not approximation: Jäckel's seminal "Let's Be Rational" (LBR) solver already inverts the Black-Scholes price to machine precision efficiently. What is missing is a differentiable layer that preserves LBR in the forward pass and avoids backpropagating through its branch logic. Such a layer must also confront the unavoidable singularity of the inverse map in the low-vega regime, where the sensitivity 1/vega diverges as vega -> 0. We close this gap with PIVOT, the Price-Implied-Volatility Objective Translator. PIVOT keeps the LBR forward pass intact and supplies the backward pass by implicit differentiation through the smooth Black-Scholes/Black-76 price map, with an explicit gating contract: invalid domains return NaN, well-conditioned rows receive the exact 1/vega gradient, and low-vega rows are attenuated rather than silently regularized. On a single H100, a fused Triton kernel reaches 1.79e9 IV/s at machine precision (9.3e-14 max relative error vs. the reference C solver); end-to-end label generation sustains 48.9M/s on synthetic chains and 16.6M/s on SPX OptionMetrics. In a HyperIV-style one-day reproduction on SPX, PIVOT-augmented objectives Pareto-dominate the baselines, reducing held-out price MAE by up to 43.4% and the strongest three-seed gated objective improving price MAE by 38.8% and IV MAE by 21.3% jointly; cross-asset results on RUT, VIX, and NDX show directional price-MAE gains of 40.1%, 24.2%, and 16.7%, while an ungated IV-roundtrip control collapses to a degenerate near-zero surface, confirming the gate as a correctness contract rather than a tuning knob.

We present a novel application of Neural Networks with Local Converging Inputs (NNLCI) to improve the efficiency of existing numerical methods for pricing multi-asset options. The most concise input format for NNLCI has been introduced, offering substantial convenience and efficiency. NNLCI uses a neural network to locally correct solutions from a coarse mesh and a refined mesh (relative to the coarse one), requiring only a minimal amount of high-fidelity training data. We demonstrate this approach on cash-or-nothing options under the Black-Scholes equation in one, two, and three spatial dimensions, and on single-asset down-and-out barrier call options under the Heston stochastic-volatility model (whose pricing PDE is two-dimensional in the spot price $S$ and the instantaneous variance $v$). In each case, NNLCI reduces the root-mean-square error (RMSE) of the refined-mesh numerical solution by a factor of approximately 4-12 on test sets, even when the neural network is trained on only a small subset of parameter combinations. These results demonstrate that NNLCI significantly reduces computational requirements for high-dimensional problems in real-time options trading and risk management, offering low training costs and strong generalization ability.