The multifractional model with random exponent (MPRE) is one of the most recent fractional models which extend the fractional Brownian motion (fBm). This paper is an empirical contribution to the justification of the MPRE. Working with several FX rates between 2006 and 2016, sampled every minute, we show the statistical significance of various fractional models applied to log-prices, from the fBm to the MPRE. We propose a method to extract realized Hurst exponents from log-prices. This provides us with a series of Hurst exponents on which we can estimate different models of dynamics. In the MPRE framework, the data justify using a fractional model for the dynamic of the Hurst exponent. We estimate and interpret the value of the key parameter of this model of nested fractality, which is the Hurst exponent of the Hurst exponents
