Joint Learning of Local-Stochastic Volatility Calibration: A Projection-Constrained Neural Operator Approach

Local-stochastic volatility (LSV) models combine the marginal calibration properties of local volatility with the richer smile dynamics of stochastic-volatility models. Their calibration, however, is governed by a circular McKean–Vlasov fixed point: the leverage function determines the simulated state distribution, while the conditional variance extracted from that distribution is required to update the same leverage function. This work presents an independent Alpha Stochastic Research implementation and controlled experimental study of a projection-constrained neural operator for LSV calibration, inspired by Wang, Després, Dureau, and Buet-Golfouse (2026). Given a finite and potentially irregular bid–ask option quote book together with a stochastic-volatility backbone descriptor, the implemented operator jointly reconstructs four linked fields: - log implied variance;- Dupire local variance;- the conditional variance moment;- squared LSV leverage. The outputs are coupled through quote consistency, static no-arbitrage margins, a division-free Dupire residual, and the Markovian-projection identity: ℓ²(t, x) × m(t, x) = a_D(t, x). The study begins with a deterministic constant-variance backbone. In this controlled setting, the conditional moment and leverage are available exactly, allowing approximation, differentiation, and optimization errors to be examined separately from particle noise and stochastic fixed-point error. The associated implementation includes: - forward-normalized Black pricing and implied-volatility inversion;- synthetic arbitrage-screened option surfaces;- Dupire local-variance reconstruction;- a permutation-invariant Deep Sets quote encoder;- four coordinate-aware DeepONet output heads;- quote-fit, no-arbitrage, Dupire, and projection losses;- an independent reference-flux diagnostic;- descriptor-space out-of-distribution screening;- an illustrative ACCEPT / POLISH / REJECT decision policy;- a complete Jupyter notebook, Python modules, tests, figures, and experiment configurations. The paper also studies how approximation error propagates from sparse implied-volatility reconstruction through Dupire differentiation and leverage recovery. The results demonstrate a complete research pipeline, but they do not establish production-level calibration accuracy. This work does not implement the full stochastic Heston-LSV particle fixed point, historical option-book validation, exotic-pricing validation, or a production particle-polishing stage. These components are identified as future research directions. This is an independent research implementation. It is not the official implementation of Wang et al., does not reproduce their unpublished infrastructure or data, and does not claim ownership of the projection-consistent neural-operator methodology introduced in their study.