From rough to multifractal multidimensional volatility: A multidimensional Log S-fBM model
Abstract
We introduce the multivariate Log S-fBM model (mLog S-fBM), extending the univariate framework proposed by Wu et al. [32] to the multidimensional setting. We begin by defining the multidimensional Stationary fractional Brownian motion (mS-fBM). This process is characterized by marginals following S-fBM dynamics and a specific cross-covariance structure. It is parametrized by a correlation scale T , marginal-specific intermittency parameters (λ 2 i ) 1≤i≤d and Hurst exponents (H i ) 1≤i≤d , as well as their multidimensional counterparts: the co-intermittency matrix (ξ i,j ) 1≤i,j≤d and the co-Hurst matrix (H i,j ) 1≤i,j≤d . The mLog S-fBM is then constructed by modeling the volatility components as exponentials of the mS-fBM, thereby preserving the dependence structure encoded in the Gaussian core. We establish that the mLog S-fBM is well-defined for any co-Hurst matrix with entries in [0, 1/2). Notably, the model supports vanishing co-Hurst parameters, allowing it to bridge the rough volatility and multifractal regimes. Furthermore, we generalize the small intermittency approximation technique to the multivariate setting to develop an efficient Generalized Method of Moments (GMM) calibration procedure. This method relies on estimating the cross-covariance parameters for pairs of marginals. We validate the procedure on synthetic data and subsequently apply it to S&P 500 market data, modeling stock return fluctuations within this multivariate stochastic volatility framework. In particular, diagonal estimates of the stock Hurst matrix, corresponding to single-stock log-volatility Hurst exponents, are consistently found to be close to H ii ≈ 0, indicating a multifractal behavior in agreement with the findings of [32]. Conversely, we observe that the co-Hurst matrix off-diagonal entries are close to the Hurst exponent of the S&P 500 index itself (H ≈ 0.15), consistent with the rough volatility literature, while the off-diagonal entries of the co-intermittency matrix align closely with univariate intermittency estimates.
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BibTeX
@misc{zarhali2026from,
title = {{From rough to multifractal multidimensional volatility: A multidimensional Log S-fBM model}},
author = {Othmane Zarhali and Emmanuel Bacry and Jean-François Muzy},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05841179v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From rough to multifractal multidimensional volatility: A multidimensional Log S-fBM model
AU - Othmane Zarhali
AU - Emmanuel Bacry
AU - Jean-François Muzy
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05841179v1
AB - We introduce the multivariate Log S-fBM model (mLog S-fBM), extending the univariate framework proposed by Wu et al. [32] to the multidimensional setting. We begin by defining the multidimensional Stationary fractional Brownian motion (mS-fBM). This process is characterized by marginals following S-fBM dynamics and a specific cross-covariance structure. It is parametrized by a correlation scale T , marginal-specific intermittency parameters (λ 2 i ) 1≤i≤d and Hurst exponents (H i ) 1≤i≤d , as well as their multidimensional counterparts: the co-intermittency matrix (ξ i,j ) 1≤i,j≤d and the co-Hurst matrix (H i,j ) 1≤i,j≤d . The mLog S-fBM is then constructed by modeling the volatility components as exponentials of the mS-fBM, thereby preserving the dependence structure encoded in the Gaussian core. We establish that the mLog S-fBM is well-defined for any co-Hurst matrix with entries in [0, 1/2). Notably, the model supports vanishing co-Hurst parameters, allowing it to bridge the rough volatility and multifractal regimes. Furthermore, we generalize the small intermittency approximation technique to the multivariate setting to develop an efficient Generalized Method of Moments (GMM) calibration procedure. This method relies on estimating the cross-covariance parameters for pairs of marginals. We validate the procedure on synthetic data and subsequently apply it to S&P 500 market data, modeling stock return fluctuations within this multivariate stochastic volatility framework. In particular, diagonal estimates of the stock Hurst matrix, corresponding to single-stock log-volatility Hurst exponents, are consistently found to be close to H ii ≈ 0, indicating a multifractal behavior in agreement with the findings of [32]. Conversely, we observe that the co-Hurst matrix off-diagonal entries are close to the Hurst exponent of the S&P 500 index itself (H ≈ 0.15), consistent with the rough volatility literature, while the off-diagonal entries of the co-intermittency matrix align closely with univariate intermittency estimates.
ER -