Published on Mon Jan 24 2022

From Rough to Multifractal volatility: the log S-fBM model

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We introduce a family of random measures $M_{H,T}(dt)$, namely log S-fBM, such that, for $H>0$, $M_{H,T}(dt)=e_{ω_{H,T}(t)}dt$ where $ω_{H,T}(t)$ is a Gaussian process that can be considered as a stationary version of a $H$-fractional Brownian motion. Moreover, when $H→0$, one has $M_{H,T}(dt)→M~_{T}(dt)$ (in the weak sense) where $M~_{T}(dt)$ is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal ($H=0$) and rough volatility ($0<H<1/2$) models. The main properties of the log S-fBM are discussed and their estimation issues are addressed. We notably show that the direct estimation of $H$ from the scaling properties of $ln(M_{H,T}([t,t+τ]))$, at fixed $τ$, can lead to strongly over-estimating the value of $H$. We propose a better GMM estimation method which is shown to be valid in the high-frequency asymptotic regime. When applied to a large set of empirical volatility data, we observe that stock indices have values around $H=0.1$ while individual stocks are characterized by values of $H$ that can be very close to $0$ and thus well described by a MRM. We also bring evidence that, unlike the log-volatility variance $ν_{2}$ whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called "intermittency coefficient" $λ_{2}$, which is the product of $ν_{2}$ and the Hurst exponent $H$, appears to be far more reliable leading to values that seems to be universal for respectively all individual stocks and all stock indices.

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