Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes

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.

We study the reconstruction of implied volatility surfaces from sparse and noisy option quotes using deep learning models under no-arbitrage constraints. We compare multiple neural architectures, including multilayer perceptrons, convolutional networks, U-Nets, variational autoencoders, and Transformer-based models against classical SVI parameterizations on option market data. Results show that Transformer and U-Net architectures achieve strong reconstruction accuracy, particularly under sparse observation regimes, while soft arbitrage penalties significantly reduce arbitrage violations with moderate impact on reconstruction error. We further analyze the trade-off between accuracy and arbitrage consistency across architectures and regularization strengths.

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.