From Rough to Multifractal volatility: the log S-fBM model
We introduce a family of random measures MH,T(dt), namely log S-fBM, such that, for H>0, MH,T(dt)=eωH,T(t)dt where ωH,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 MH,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(MH,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 ν2 whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called "intermittency coefficient" λ2, which is the product of ν2 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.