From rough to multifractal multidimensional volatility: A multidimensional Log S-fBM model

We introduce the multivariate Log S-fBM model (mLog S-fBM), extending the univariate framework proposed by Wu \textit{et al.} to the multidimensional setting. We define the multidimensional Stationary fractional Brownian motion (mS-fBM), 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 and Hurst exponents, as well as their multidimensional counterparts: the co-intermittency matrix and the co-Hurst matrix. The mLog S-fBM is constructed by modeling volatility components as exponentials of the mS-fBM, preserving the dependence structure of the Gaussian core. We demonstrate that the model is well-defined for any co-Hurst matrix with entries in $[0, \frac{1}{2}[$, supporting vanishing co-Hurst parameters to bridge rough volatility and multifractal regimes. We generalize the small intermittency approximation technique to the multivariate setting to develop an efficient Generalized Method of Moments calibration procedure, estimating cross-covariance parameters for pairs of marginals. We validate it on synthetic data and apply it to S\&P 500 market data, modeling stock return fluctuations. Diagonal estimates of the stock Hurst matrix, corresponding to single-stock log-volatility Hurst exponents, are close to 0, indicating multifractal behavior, while co-Hurst off-diagonal entries are close to the Hurst exponent of the S\&P 500 index ($H \approx 0.12$), and co-intermittency off-diagonal entries align with univariate intermittency estimates.