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 to modeling them properly . The Levy'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 fairly well by a Levy's stable distribution with alpha = 1.4 , which indicates fatter tails compared to a Gaussian distribution. Moreover, we have come to realize the limitations of the stable model especially under certain conditions outside time intervals ranging  from 30 minutes to 4 hours. The arguments are developed under the theoretical background of the General Central Limit Theorem (GCLT) and quantitative valuation defined as a distance function using the Parseval's relation . Our approach can generally be extended for further analysis of statistical properties and contribute to develop proper applications for financial modeling.
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 Levy'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 Levy's stable distribution with \alpha \simeq 1.4 under certain conditions on time intervals ranging roughly from 30 minutes to 4 hours. 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 fit with the Levy's stable distribution compared to the fit with other distributions by employing the method based on likelihood ratios . Our approach can be extended for further analysis of statistical properties and contribute to developing proper applications for financial modeling.
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