# Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility

arXiv: https://arxiv.org/abs/2608.01217 · HTML: https://mlsanity.com/paper/2608.01217/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [A generative adversarial network approach to calibration of local stochastic volatility models](https://mlsanity.com/paper/2005.02505/) (2020)
- [Deep Smoothing of the Implied Volatility Surface](https://mlsanity.com/paper/1906.05065/) (2019)
- [Deep calibration of rough stochastic volatility models](https://mlsanity.com/paper/1810.03399/) (2018)
- [Differential Machine Learning](https://mlsanity.com/paper/2005.02347/) (2020)

## Abstract (arXiv)

Local-stochastic volatility (LSV) combines vanilla marginals with richer smile dynamics, but calibration requires a slow, noisy and sequential McKean--Vlasov fixed point. We learn a projection-consistent operator for the calibration triple. Given finite quotes and a stochastic-volatility (SV) backbone, it jointly returns an implied-volatility surface subject to static-arbitrage constraints, its Dupire local volatility, LSV leverage and the conditional moment required by the projection identity. Starting from option-price marginals, we derive a division-free Dupire residual in log-implied-variance coordinates and a quotient Fokker--Planck equation after Gy"ongy projection. Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) implementations enforce quote fit, static-arbitrage, Dupire and projection constraints. For the witness-augmented residual system, we prove conditional identification and empirical consistency under LSV existence and inverse residual stability. In controlled synthetic tests, forward-start and cliquet errors differ from a particle method by 0.1 and 0.2 percentage points, while calibration latency falls from 98.5 to 0.6 ms. Compared with the tested baselines, local-volatility root-mean-square error (RMSE) falls by 36% and leverage RMSE by 7-16%. These results support amortizing the LSV fixed point: the expensive solve moves offline, while online calibration reduces to a single projection-consistent operator evaluation.