A regularity structure for rough volatility

Abstract

A new paradigm has emerged recently in financial model-ing: rough (stochastic) volatility. First observed by Gatheralet al. in high-frequency data, subsequently derived withinmarket microstructure models, rough volatility capturesparsimoniously key-stylized facts of the entire impliedvolatility surface, including extreme skews (as observedearlier by Alòs et al.) that were thought to be outside thescope of stochastic volatility models. On the mathematicalside, Markovianity and, partially, semimartingality are lost.In this paper, we show that Hairer’s regularity structures, amajor extension of rough path theory, which caused a rev-olution in the field of stochastic partial differential equa-tions, also provide a new and powerful tool to analyze roughvolatility models.