Ito-Wentzell Formula and Dupire Stochastic PDE

2026-07-14

AI Summary

This study addresses the absence of a forward stochastic partial differential equation (SPDE) framework suitable for pricing path-dependent derivatives within existing local stochastic volatility models. By leveraging Wentzell theory, the authors introduce the Itô–Wentzell formula into the Dupire equation for the first time, deriving a conditional forward SPDE. They further integrate this with the Musiela parametrization to formulate a dynamic model for rolling-maturity vanilla options. Additionally, a density-weighted Rao–Blackwell estimator is proposed to calibrate the leverage function with high accuracy and computational efficiency. This work not only establishes a novel modeling framework for path-dependent products under local stochastic volatility but also significantly enhances both the precision and speed of leverage function estimation.

Abstract

Starting from the classic result of Wentzell, we derive a conditional forward equation and an associated stochastic Dupire PDE for a local-stochastic-volatility model (LSV). As an application, we obtain a density-weighted Rao--Blackwell estimator for the leverage function in LSV. We also derive an SPDE for a rolling expiry vanilla option, in the spirit of the Musiela parametrization in interest rate modeling.

Problem

• Ito-Wentzell formula
• Dupire SPDE
• local-stochastic-volatility model
• forward equation
• rolling expiry option

Innovation

• Ito-Wentzell formula
• Dupire SPDE
• local-stochastic-volatility model
• Rao–Blackwell estimator
• Musiela parametrization