Scale-Dependent Multifractality in Bitcoin Realised Volatility: Implications for Rough Volatility Modelling

Abstract: We assess the applicability of rough volatility models to Bitcoin realised volatility using the normalised p-variation framework of Cont and Das (2024). Applying this model free estimator to high-frequency Bitcoin data from 2017 to 2024 across multiple sampling resolutions, we find that the normalised statistic remains strictly negative throughout, precluding the estimation of a valid roughness index. Stationarity tests and robustness checks reveal no significant evidence of non-stationarity or structural breaks as explanatory factors. Instead, convergent evidence from three complementary diagnostics, namely multifractal detrended fluctuation analysis, log-log moment scaling, and wavelet leaders, reveals a multifractal structure in Bitcoin volatility. This scale-dependent behaviour violates the homogeneity assumptions underlying rough volatility estimation and accounts for the estimator's systematic failure. These findings suggest that while rough volatility models perform well in traditional markets, they are structurally misaligned with the empirical features of Bitcoin volatility.