Log-Modulated Rough Stochastic Volatility Models

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The so-obtained logmodulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting super-rough stochastic volatility models can be analyzed over the whole range 0 <= H < 1/2 without the need of further normalization. We obtain skew asymptotics of the form log(1/T)(-pT H 1/2) as T -> 0, H >= 0, so no flattening of the skew occurs as H -> 0.