From rough to multifractal volatility: The log S-fBM model
Résumé
We introduce a family of random measures MH,T (dt), namely log S-fBM, such that, for H > 0, MH,T (dt) = e ω H,T (t) dt where ωH,T (t) is a Gaussian process that can be considered as a stationary version of an H-fractional Brownian motion. Moreover, when H → 0, one has MH,T (dt) → MT (dt) (in the weak sense) where MT (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(MH,T ([t, t+τ ])), at fixed τ , can lead to strongly overestimating 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 seem to be universal for respectively all individual stocks and all stock indices.
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