  We investigate the behaviour of cryptocurrencies using data for bitcoin,
ethereum and ripple which account for over 70% of the cryptocurrency market. We
demonstrate that $\alpha$-stable distribution is an appropriately sufficient
model for highly speculative cryptocurrencies which outperforms other heavy
tailed distributions that are used in financial econometrics. We find that the
maximum likelihood method proposed by DuMouchel (1971) produces estimates that
fit the cryptocurrency return data much better than the quantile based approach
of McCulloch (1986) and sample characteristic method by Koutrouvelis (1980).
The empirical results show that the leptokurtic feature presented in
cryptocurrency return data can be captured by an $\alpha$-stable distribution.
The findings highlight that $\alpha$-stable distribution is not only
parsimonious with its four free parameters but also a creative model that is
close to reality. This paper covers early reports and literature on
cryptocurrencies and stable distributions.
