Precise asymptotics: Robust stochastic volatility models

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance 30 (2020) 782-832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama-Kawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance 16 (2016) 887-904) and Forde-Zhang (SIAM J. Financial Math. 8 (2017) 114-145), one obtains precise asymptotics for European options which refine known large deviation asymptotics.