Characterizing Cryptocurrency Market with Lévy’s Stable Distributions

Abstract

The recent emergence of cryptocurrencies such as Bitcoin and Ethereum has posed possible alternatives to global payments as well as financial assets around the globe, making investors and financial regulators aware of the importance of modeling them correctly. The Lévy’s stable distribution is one of the attractive distributions that well describes the fat tails and scaling phenomena in economic systems. In this paper, we show that the behaviors of price fluctuations in emerging cryptocurrency markets can be characterized by a non-Gaussian Lévy’s stable distribution with α≃1.4 under certain conditions on time intervals ranging roughly from 30 min to 4 h. Our arguments are developed under quantitative valuation defined as a distance function using the Parseval’s relation in addition to the theoretical background of the General Central Limit Theorem (GCLT). We also discuss the model-fitting for returns by employing the method based on likelihood ratios. Even though the cubic power-law model is a better fitting model than the Lévy’s stable model in the tail part of returns, the Lévy’s stable model outperforms the fit for the entire and wider range of returns. Our approach can be extended for further analysis of statistical properties and contribute to developing proper applications for financial modeling.