Modelling the cryptocurrency return distribution via Laplace scale mixtures

The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector θ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of θ. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution.