We develop a variant of rough path theory tailor-made for the analysis of a class of financial asset price models, the so-called 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 behaviour 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, an LDP on the partial rough path space is proved and the LDP for rough volatility then follows by the continuity of the solution map of rough differential equations (RDE). This is a joint work with Ryoji Takano.
Masaaki Fukasawa is Professor of Mathematics in the Graduate School of Engineering Science at Osaka University.
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