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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 Lévy's stable distribution is one of the attractive distribution that well describes the fat tails and scaling phenomena in economic systems. In this paper, we show that the behaviors of price fluctuations in cryptocurrency markets can also be characterized by a Lévy'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.

This article investigates the dynamical complexity and fractal characteristics changes of the Bitcoin/US dollar (BTC/USD) and Euro/US dollar (EUR/USD) returns in the period before and after the outbreak of the COVID-19 pandemic. More specifically, we applied the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) method to investigate the temporal evolution of the asymmetric multifractal spectrum parameters. In addition, we examined the temporal evolution of Fuzzy entropy, non-extensive Tsallis entropy, Shannon entropy, and Fisher information. Our research was motivated to contribute to the comprehension of the pandemic’s impact and the possible changes it caused in two currencies that play a key role in the modern financial system. Our results revealed that for the overall trend both before and after the outbreak of the pandemic, the BTC/USD returns exhibited persistent behavior while the EUR/USD returns exhibited anti-persistent behavior. Additionally, after the outbreak of COVID-19, there was an increase in the degree of multifractality, a dominance of large fluctuations, as well as a sharp decrease of the complexity (i.e., increase of the order and information content and decrease of randomness) of both BTC/USD and EUR/USD returns. The World Health Organization (WHO) announcement, in which COVID-19 was declared a global pandemic, appears to have had a significant impact on the sudden change in complexity. Our findings can help both investors and risk managers, as well as policymakers, to formulate a comprehensive response to the occurrence of such external events.

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