Ever smaller Hurst parameter estimates of the fractional Brownian motion (fBm) underlying to rough volatility model motivated Neuman and Rosenbaum (2017) to treat the convergence of a fBm with vanishing Hurst parameter. Due to the vanishing Hölder regularity, the convergence can only be meaningful in the sense of generalised functions. The authors posed a normalisation of fBm which convergences non-degenerate to a log-correlated process, i.e. a Gaussian process with a logarithmic singularity on the diagonal of the covariance kernel. Placed in the abstract ambient of the less known fractional Gaussian fields, we find the essential property of the normalisation that makes this convergence possible and we pose a general class of normalisations, which apply to fractional Brownian fields as well.
The more interesting object from the finance perspective is the volatility process, i.e the exponential of the fBm. The limit process is not point-wise defined, yet we can make sense of the integrated volatility process as a random measure. The theory of Gaussian multiplicative chaos deals exactly with the latter problem: Defining the exponential of singular Gaussian fields. The usual construction of these measures uses a regularisation of the field, therefore most results are not directly applicable in our situation. We are able to state an elementary proof for the convergence of the volatility measures in the so call L2-phase.