Log-modulated rough stochastic volatility models

TL;DR

This paper introduces a new class of rough stochastic volatility models using log-modulated fractional Brownian motion, enabling analysis across the entire roughness spectrum without normalization, and deriving specific skew asymptotics.

Contribution

It proposes a novel log-modulated fractional Brownian motion framework that remains well-defined for all Hurst indices in [0, 1/2), expanding the scope of rough volatility modeling.

Findings

Models are valid for H in [0, 1/2) without normalization.

Derived skew asymptotics show no flattening as H approaches 0.

Log-modulated fBm retains Gaussian properties for H=0.

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 $H \to 0$.