Before revision:  In this paper we first show that the behaviors of price fluctuations of cryptocurrencies can also be characterized by the fat-tail Levy's stable distribution by our parameter estimation method. After confirming that price returns of cryptocurrencies follow Levy's stable distribution, we discuss the recent market instability by focusing on one of the parameters of the distribution, which can characterize the fat-tailed behavior of cryptocurrency price returns. Our analysis shows that the fluctuations of estimated tail index parameter could be a candidate for the measure to capture extreme price behaviors of recently emerging cryptocurrencies and the tail index can be applicable for risk management and their financial modeling.
After revision:  In this paper we first show that the behaviors of price fluctuations of cryptocurrencies can also be characterized by the fat-tail Levy's stable distribution under certain conditions of 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 applications for financial modeling.