From rough to multifractal volatility: The log S-fBM model

We introduce a family of random measures $M_{H,T} (d t)$, namely log S-fBM, such that, for $H>0$, $M_{H,T}(d t) = e^{\omega_{H,T}(t)} d t$ where $\omega_{H,T}(t)$ is Gaussian process that can be considered as stationary version an $H$-fractional Brownian motion. Moreover, when $H \to 0$, one has \rightarrow {\widetilde M}_{T}(d t)$ (in the weak sense) ${\widetilde celebrated log-normal multifractal measure (MRM). Thus, this model allows us to consider, within same framework, two popular classes ($H 0$) and rough volatility ($0<H < 1/2$) models. The main properties S-fBM are discussed their estimation issues addressed. notably show direct $H$ from scaling $\ln(M_{H,T}([t, t+\tau]))$, at fixed $\tau$, lead strongly over-estimating value $H$. propose better GMM method which shown valid in high-frequency asymptotic regime. When applied large set empirical data, we observe stock indices have values around $H=0.1$ while individual stocks characterized by very close $0$ thus well described MRM. also bring evidence unlike log-volatility variance $\nu^2$ whose appears poorly reliable (though used widely literature), so-called "intermittency coefficient" $\lambda^2$, product Hurst exponent $H$, far more leading seem universal respectively all indices.