  A new paradigm recently emerged in financial modelling: rough (stochastic)
volatility, first observed by Gatheral et al. in high-frequency data,
subsequently derived within market microstructure models, also turned out to
capture parsimoniously key stylized facts of the entire implied volatility
surface, including extreme skews that were thought to be outside the scope of
stochastic volatility. On the mathematical side, Markovianity and, partially,
semi-martingality are lost. In this paper we show that Hairer's regularity
structures, a major extension of rough path theory, which caused a revolution
in the field of stochastic partial differential equations, also provides a new
and powerful tool to analyze rough volatility models.
