A partial rough path space for rough volatility

TL;DR

This paper introduces a specialized rough path framework for rough volatility models in finance, establishing a large deviation principle that describes short-term implied volatility behavior.

Contribution

It develops a partial rough path space and an integration map, enabling the analysis of rough volatility models and deriving a pathwise large deviation principle.

Findings

Established a partial rough path space with an integration map.

Proved a large deviation principle for rough volatility models.

Described the limiting behavior of implied volatility for short maturities.

Abstract

We develop a variant of rough path theory tailor-made for analyzing a class of financial asset price models known as rough volatility models. As an application, we prove a pathwise large deviation principle (LDP) for a certain class of rough volatility models, which in turn describes the limiting behavior of implied volatility for short maturity under those models. First, we introduce a partial rough path space and an integration map on it and then investigate several fundamental properties including local Lipschitz continuity of the integration map from the partial rough path space to a rough path space. Second, we construct a rough path lift of a rough volatility model. Finally, we prove an LDP on the partial rough path space, and the LDP for rough volatility then follows by the continuity of the solution map of rough differential equations.