The Log S-fBM model: Statistical analysis
Abstract
The Log S-fBM model is a stochastic volatility model introduced by Wu et al. in [25]. It is characterized by a stochastic log volatility being a stationary fractional brownian motion (S-fBM) process: a stationary gaussian process whose autocovariance function has a power decay driven by the Hurst exponent H and variance having a multiplicative coefficient namely the intermittency coefficient. One of the main properties of the Log S-fBM model is that it conciliates the rough volatility setting where the Hurst exponent is typically near 0.1 (see [24]) and the multifractal volatility setting where the Hurst exponent is of order 0 as introduced in [23, 29] as follows: the volatility measure of the Log S-fBM model tends to the one of multifractal volatility when the Hurst exponent goes to 0. According to the numerical findings in [25], this intermittency was observed to be of order 0.02 across a large range of financial assets which justifies the meaningfulness of considering a small intermittency approximation of log volatility moments as a model calibration method (so called general method of moment known as GMM). In this work, we perform a statistical analysis of the Log S-fBM model. First, we derive scaling properties related to the S-fBM process as well as the Log S-fBM integrated volatility measure. Then, we present deviation inequalities of the Log S-fBM process with a precise description of its tail distribution emphasizing on its sensitivity with respect to the Hurst exponent and the intermittency coefficient. Moreover, we develop a statistical hypothesis testing in order to test the null hypothesis of the Hurst exponent, in other words, test the rough VS multifractal hypothesis. Last but not least, we revisit the scale invariance properties of the log volatility increment process with explicit formulas leveraging the small intermittency approximation, enabling to reproduce the analoguous properties to the rough as well as the multifractal volatility settings.
BibTeX
@misc{zarhali2026sfbm,
title = {{The Log S-fBM model: Statistical analysis}},
author = {Othmane Zarhali and Emmanuel Bacry and Jean-François Muzy},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05841180v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Log S-fBM model: Statistical analysis
AU - Othmane Zarhali
AU - Emmanuel Bacry
AU - Jean-François Muzy
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05841180v1
AB - The Log S-fBM model is a stochastic volatility model introduced by Wu et al. in [25]. It is characterized by a stochastic log volatility being a stationary fractional brownian motion (S-fBM) process: a stationary gaussian process whose autocovariance function has a power decay driven by the Hurst exponent H and variance having a multiplicative coefficient namely the intermittency coefficient. One of the main properties of the Log S-fBM model is that it conciliates the rough volatility setting where the Hurst exponent is typically near 0.1 (see [24]) and the multifractal volatility setting where the Hurst exponent is of order 0 as introduced in [23, 29] as follows: the volatility measure of the Log S-fBM model tends to the one of multifractal volatility when the Hurst exponent goes to 0. According to the numerical findings in [25], this intermittency was observed to be of order 0.02 across a large range of financial assets which justifies the meaningfulness of considering a small intermittency approximation of log volatility moments as a model calibration method (so called general method of moment known as GMM). In this work, we perform a statistical analysis of the Log S-fBM model. First, we derive scaling properties related to the S-fBM process as well as the Log S-fBM integrated volatility measure. Then, we present deviation inequalities of the Log S-fBM process with a precise description of its tail distribution emphasizing on its sensitivity with respect to the Hurst exponent and the intermittency coefficient. Moreover, we develop a statistical hypothesis testing in order to test the null hypothesis of the Hurst exponent, in other words, test the rough VS multifractal hypothesis. Last but not least, we revisit the scale invariance properties of the log volatility increment process with explicit formulas leveraging the small intermittency approximation, enabling to reproduce the analoguous properties to the rough as well as the multifractal volatility settings.
ER -