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

TL;DR

The paper extends the univariate Log S-fBM framework to a multidimensional setting, defining a multidimensional stationary fractional Brownian motion (mS-fBM) with a co-intermittency matrix $\bm{\xi}$ and a co-Hurst matrix $\bm{H}$ to capture cross-asset dependence in volatility. It constructs the mLog S-fBM by modeling volatility components as exponentials of the mS-fBM, and shows the limit $\|\bm{H}\|\to 0$ yields a multidimensional multifractal random measure, unifying rough and multifractal regimes. A small intermittency approximation is developed for the multivariate case, enabling a generalized method of moments (GMM) calibration to estimate cross-covariance parameters $(g_{i,j},H_{i,j})$ from data, with validation on synthetic data. The framework is then applied to S&P 500 data, revealing near-zero diagonal log-volatility Hurst exponents (multifractal behavior) and off-diagonal co-Hurst entries near the index Hurst level ($H\approx 0.12$), while off-diagonal intermittency alignments are consistent with univariate estimates. Overall, the model provides a parsimonious, interpretable multivariate volatility description that bridges rough and multifractal dynamics and captures cross-asset dependence across scales.

Abstract

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.

Figures (5)

• Figure 1: Representation of $C_{\ell,T}(.)$.
• Figure 2: Empirical distributions of respectively $H_{i,j}$(top) and $g_{i,j}$(bottom) from a sample of $200$ copies of each obtained from synthetically generated data as well as the corresponding theoretical values. Simulations with $H_1=H_2=0.02$, $\lambda_1^2=\lambda_2^2=0.05$, $T=N=2^{14}$ and subsampling length $2^4$.
• Figure 3: The log standard deviation of the calibrated model parameters for $H_1=H_2=0.02$, $H_{1,2}=0.15$, $T=2^{14}$ and $g_{1,2}=0.5$ as a function of log length. Here, $\lambda_1^2=\lambda_2^2=0.05$. In blue and orange the respective linear fits.
• Figure 4: Scatter plots for all the selected assets of the S&P500 of: the Hursts and co-Hurst exponents (a), the intermittency and co-intermittency coefficients (b) and the co-intermittency correlation coefficients (c).
• Figure 5: Histograms of empirical values of ${\widehat{H}}$ and ${\widehat{g}}$ from Fig. \ref{['fig:Hassets']}

Theorems & Definitions (28)

• Corollary 1
• Remark 1
• Definition 1: mS-fBM process
• Remark 2
• Proposition 1: cross covariance of the mS-fBM
• Proposition 2: cross covariance of $\Omega$
• Definition 2: mLog S-fBM random measure
• Definition 3
• Proposition 3: multifractality of $\widetilde{\bm{M}}_{T}$
• Theorem 1: The limit $\|\bm{H} \|\rightarrow 0$