An analysis of the roughness of cryptocurrency volatility

The goal of this work is to assess whether cryptocurrency volatility paths exhibit roughness in the same way as the rest of the financial instruments by using the method of absolute moments developed by Gatheral, Jaisson and Rosenbaum and the normalized p-th order variation developed by Cont and Das. The OU and the f-OU stochastic volatility models are simulated using the Euler-Maruyama scheme and the performance of the estimators mentioned above is assessed finding that, no matter the choice of the parameters of the process or the choice of the driving process of the SDE, realized volatility is always rough. Then the same analysis is replicated using respectively the Oxford-Man realized volatility data and cryptocurrency data. First the method of absolute moment is tested, finding results consistent with Gatheral, Jaisson and Rosenbaum; then the original analysis is expanded by estimating the roughness index via the normalized p-th order variation technique, again finding similar results. Indeed, cryptocurrencies show roughness as the rest of the financial ecosystem. Nonetheless the rough paradigm is critiqued as it does not accurately describe the volatility dynamics at longer time-scales and because the roughness of realized volatility paths does not constitute evidence for the roughness of the latent instantaneous volatility process.