$\mathfrak{X}$PDE for $\mathfrak{X} \in \{\mathrm{BS},\mathrm{FBS}, \mathrm{P}\}$: a rough volatility context

TL;DR

This paper establishes a precise connection between path-dependent PDEs and backward SPDEs in the context of rough volatility, clarifying the nature of a previously introduced random field as a pathwise derivative of the value function.

Contribution

It explicitly links path-dependent PDEs and backward SPDEs in rough volatility models, identifying the random field as a pathwise derivative of the value function.

Findings

Clarified the connection between path-dependent PDEs and backward SPDEs in rough volatility.

Identified the random field as a pathwise derivative of the value function.

Enhanced understanding of mathematical structures in rough volatility models.

Abstract

Recent mathematical advances in the context of rough volatility have highlighted interesting and intricate connections between path-dependent partial differential equations and backward stochastic partial differential equations. In this note, we make this link precise, identifying the slightly obscure random field introduced in [Pricing options under rough volatility with backward SPDEs, C. Bayer; J. Qiu and Y. Yao. SIFIN, 13(1), 179-212 (2022)] as a pathwise derivative of the value function.