arXiv subjects

Othmane Zarhali

Publications and source records attributed to Othmane Zarhali.

4 recordsLinked to original sources

arXiv2026

The Log S-fBM model: Statistical analysis

The Log S-fBM model, introduced by Wu et al., is a stochastic volatility model whose log volatility is a stationary fractional Brownian motion (S-fBM): a stationary Gaussian process with power-decaying autocovariance driven by the Hurst exponent $H$, and variance scaled by an intermittency coefficient. A key property is that it reconciles rough volatility, where $H$ is typically near $0.1$ (see Gatheral et al.), with multifractal volatility, where $H$ is close to $0$ as in Bacry, Muzy et al.: the model's volatility measure converges to a multifractal random measure as $H\to0$. Numerical findings in Wu et al. show intermittency of order $0.02$ across financial assets, motivating a small intermittency approximation of log volatility moments for calibration via the general method of moments (GMM). In this work, we conduct a statistical analysis of the Log S-fBM model. We derive scaling properties of the S-fBM process and the Log S-fBM integrated volatility measure, present deviation inequalities with tail distributions sensitive to $H$ and intermittency, and develop a hypothesis test for the null Hurst exponent, i.e.\ rough versus multifractal dynamics. Finally, we revisit scale invariance of the log volatility increment process via explicit small-intermittency formulas, reproducing analogous properties in both regimes.

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arXiv2026

A Nested Factor Model for Equity Markets: Reconciling Multifractal Stock Returns and Rough Index Volatilities

The Nested factor model was introduced by Chicheportiche et al. to represent non-linear correlations between stocks. Stock returns are explained by a standard factor model, but the (log)-volatilities of factors and residuals are themselves decomposed into factor modes, with a common dominant volatility mode affecting both market and sector factors but also residuals. Here, we consider the case of a single factor where the only dominant log-volatility mode is rough, with a Hurst exponent $H \simeq 0.11$ and the log-volatility residuals are ''super-rough'' or ''multifractal'', with $H \simeq 0$. We demonstrate that such a construction naturally accounts for the somewhat surprising stylized fact reported by Wu et al. , where it has been observed that the Hurst exponents of stock indexes are large compared to those of individual stocks. We propose a statistical procedure to estimate the Hurst factor exponent from the stock returns dynamics together with theoretical guarantees of its consistency. We demonstrate the effectiveness of our approach through numerical experiments and apply it to daily stock data from the S&P500 index. The estimated roughness exponents for both the factor and idiosyncratic components validate the assumptions underlying our model.

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arXiv2026

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.

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arXiv2026

Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion

A fast simulation framework for stochastic Volterra processes based on Random Fourier Features (RFF) approximation of the kernel is developed. After recalling the main properties of Volterra processes and reviewing existing numerical simulation methods, an accelerated scheme is introduced that relies on a spectral representation of the kernel. A particular attention is devoted to sampling from the kernel spectral density using Hamiltonian Monte Carlo, whose efficiency and stability bring more convenience than alternative sampling procedures. Quantitative guarantees for the proposed method are established, including moment estimates and strong error bounds. The approach is further compared with the kernel approximation by sum of exponentials commonly used in the literature, emphasizing the broader generality of the present framework. As a primary application, Volterra processes associated with the Stationary fractional Brownian Motion (S-fBM) kernel are investigated. A spectral density representation is derived in closed form using hypergeometric functions, a condition for positive definiteness is established and explicit truncation as well as Monte Carlo error bounds are provided for the RFF approximation in this setting. Numerical experiments in dimensions one and two illustrate the accuracy of the kernel approximation, the reliable recovery of model parameters and the competitiveness of the accelerated simulation scheme in terms of computational efficiency and both weak and strong error performance.

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