  Pricing derivatives goes back to the acclaimed Black and Scholes model.
However, such a modeling approach is known not to be able to reproduce some of
the financial stylized facts, including the dynamics of volatility. In the
mathematical finance community, it has therefore emerged a new paradigm, named
rough volatility modeling, that represents the volatility dynamics of financial
assets as a fractional Brownian motion with Hurst exponent very small, which
indeed produces rough paths. At the same time, prices' time series have been
shown to be multiscaling, characterized by different Hurst scaling exponents.
This paper assesses the interplay, if present, between price multiscaling and
volatility roughness, defined as the (low) Hurst exponent of the volatility
process. In particular, we perform extensive simulation experiments by using
one of the leading rough volatility models present in the literature, the rough
Bergomi model. A real data analysis is also carried out in order to test if the
rough volatility model reproduces the same relationship. We find that the model
is able to reproduce multiscaling features of the prices' time series when a
low value of the Hurst exponent is used but it fails to reproduce what the real
data say. Indeed, we find that the dependency between prices' multiscaling and
the Hurst exponent of the volatility process is diametrically opposite to what
we find in real data, namely a negative interplay between the two.
