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

2026-08-02

AI Summary

This work addresses the computational expense, noise sensitivity, and iterative McKean–Vlasov fixed-point solving inherent in calibrating local stochastic volatility (LSV) models by introducing the first end-to-end neural operator framework. The proposed method jointly outputs arbitrage-free implied volatility surfaces, Dupire local volatilities, LSV leverage functions, and projection-consistent conditional moments. By shifting fixed-point computation to an offline phase, online calibration requires only a single operator evaluation. The approach innovatively incorporates a division-free Dupire residual and a quotient-form Fokker–Planck equation, combining DeepONet with Fourier neural operators to enforce joint constraints—on market quote fitting, static no-arbitrage conditions, dynamic consistency, and projection identities—in log-implied-variance coordinates. Experiments demonstrate a reduction in calibration latency from 98.5 ms to 0.6 ms, a 36% decrease in local volatility RMSE, 7–16% lower leverage function RMSE, and forward-start and cliquet option pricing errors of merely 0.1% and 0.2%, respectively.

Abstract

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 Gyöngy 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.

Problem

• Local-Stochastic Volatility
• Calibration
• Projection-Consistency
• Static Arbitrage
• Dupire Equation

Innovation

• projection-consistent neural operator
• local-stochastic volatility
• amortized calibration
• Dupire equation
• Deep Operator Network