Investigating long and short memory in cryptocurrency time series by stochastic fractional Brownian models

Over the last years, the world of cryptocurrencies has undergone a tumultuous development, mostly characterized by speculative behaviors, and thus one might argue that it does not satisfy the Efficient Market Hypothesis. Since the efficiency of a financial market can be assessed by checking whether the times series of the assets traded on it are persistent/anti-persistent, we investigate the presence of memory in the price of seven among the most important cryptocurrencies. To this aim, we employ an original approach based on two fractional models, namely the geometric fractional Brownian motion and the geometric mixed fractional Brownian motion, which are tested against the Markovian geometric Brownian motion to detect the presence of memory. The above fractional models are estimated by employing an innovative maximum likelihood procedure that exploits the Toeplitz structure of the covariance matrix of log-returns. The null assumption of absence of memory in the time series is tested based on confidence ellipses for the estimated parameters. We validate the proposed procedure on artificial data and we apply it to the historical series of crypto-prices. Results suggest the presence of memory effects only for some of the considered cryptocurrencies. A possible explanation of such a persistence/anti-persistence phenomenon is provided.