Log-modulated rough stochastic volatility models

Abstract

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 log-modulated 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 analysed over the whole range $0\le H<1/2$ without the need of further normalization. We obtain skew asymptotics of the form $\log (1/T{)}^{-p}{T}^{H-1/2}$ as $T\to 0$, $H\ge 0$, so no flattening of the skew occurs as $H\to 0$.

arXiv comment28 pages, 9 figures