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From rough to multifractal volatility: Topics around the Log S-fBM model
Abstract
The Log Stationary Fractional Brownian Motion(LogS-fBM)model, introduced by Peng, Bacry, and Muzy , describes a log-volatility process driven by a stationary fractional Brownian motion (S-fBM). This model is characterized by three key parameters: the intermittency parameter λ, the correlation scale T, and the Hurst exponent H. Notably, as H approaches zero, the model’s multifractal random measure (volatility measure) converges to that of the multifractal random walk introduced by Bacry et al.. In contrast, when H ≈0.1, the model captures rough volatility dynamics.
A multidimensional extension of the Log S-fBM model, referred to as the m-Log S-fBM was also developed. In this framework, the log-volatilities of multiple assets are correlated, with dependencies governed by both the cointermittency matrix and the coHurst matrix. These matrices ensure that the marginal distributions of the model retain the one-dimensional Log S-fBM dynamic. A key analytical tool for studying this model is the small intermittency approximation, which allows to approximate the generalized moments of the normalized log-volatility over a time period ∆ > 0 using the moments of the integrated S-fBM process over the same period when λ^2 is small. This approximation is particularly relevant given the empirical findings of Wu et al., who observed that for various assets, λ^2 ≈0.02. Besides, the Log S-fBM model can be used in the Nested factor model, introduced by Bouchaud et al., where the asset return fluctuations are explained by common factors representing the market economic sectors and residuals (noises). These residuals share with the factors a common dominant volatility mode in addition to the idiosyncratic mode unique to each residual. Here, we consider the case of a single factor, where the only dominant common mode is a S-fBM process with Hurst exponent H ≃0.11, while the residuals, in addition to the previous common mode, contain idiosyncratic components with Hurst exponents H ≃0. Furthermore, we propose a statistical procedure to estimate the Hurst factor exponent from stock return dynamics, providing theoretical guarantees. The method performs well in the limit where the number of stocks N tends to infinity. In this talk, we introduce the Log S-fBM model in its one-dimensional and multidimensional forms, present the calibration procedure based on the small intermittency approximation, and discuss the Nested Log S-fBM factor model.
