{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T05:00:03Z","timestamp":1778648403724,"version":"3.51.4"},"reference-count":131,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T00:00:00Z","timestamp":1626048000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017\u20132020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called \u201cinverse-cubic power-law\u201d still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors\u2014speed of the market time flow and the asset cross-correlation magnitude\u2014while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.<\/jats:p>","DOI":"10.3390\/e23070884","type":"journal-article","created":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T21:56:37Z","timestamp":1626126997000},"page":"884","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Financial Return Distributions: Past, Present, and COVID-19"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2131-7440","authenticated-orcid":false,"given":"Marcin","family":"W\u0105torek","sequence":"first","affiliation":[{"name":"Faculty of Computer Science and Telecommunications, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8813-9637","authenticated-orcid":false,"given":"Jaros\u0142aw","family":"Kwapie\u0144","sequence":"additional","affiliation":[{"name":"Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1613-6175","authenticated-orcid":false,"given":"Stanis\u0142aw","family":"Dro\u017cd\u017c","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Telecommunications, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak\u00f3w, Poland"},{"name":"Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.24033\/asens.476","article-title":"Th\u00e9orie de sp\u00e9culation","volume":"3","author":"Bachelier","year":"1900","journal-title":"Ann. Sci. l\u2019Ecole Norm. Sup\u00e9r."},{"key":"ref_2","unstructured":"L\u00e9vy, P. (1925). Calcul des Probabilit\u00e9s, Gauthier-Villars."},{"key":"ref_3","unstructured":"Paul, W., and Baschnagel, J. (1999). Stochastic Processes: From Physics to Finance, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1086\/294632","article-title":"The variation of certain speculative prices","volume":"36","author":"Mandelbrot","year":"1963","journal-title":"J. Bus."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1086\/294743","article-title":"The behavior of stock-market prices","volume":"38","author":"Fama","year":"1965","journal-title":"J. Bus."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1086\/295262","article-title":"Portfolio theory: A step towards its practical application","volume":"43","author":"Blume","year":"1970","journal-title":"J. Bus."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1080\/01621459.1971.10482254","article-title":"A note on the distribution of stock price changes","volume":"66","author":"Teichmoeller","year":"1970","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1086\/295634","article-title":"A comparison of the stable and Student distributions as statistical models for stock prices","volume":"47","author":"Blattberg","year":"1974","journal-title":"J. Bus."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1080\/01621459.1972.10481297","article-title":"The distribution of stock returns","volume":"67","author":"Officer","year":"1972","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1080\/01621459.1973.10482431","article-title":"A reexamination of the empirical distribution of stock price changes","volume":"68","author":"Barnea","year":"1973","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/BF01096940","article-title":"Real estate is not normal: A fresh look at real estate return distributions","volume":"10","author":"Young","year":"1995","journal-title":"J. Real Estate Financ. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"135","DOI":"10.2307\/1913889","article-title":"A subordinated stochastic process model with finite variance for speculative prices","volume":"41","author":"Clark","year":"1973","journal-title":"Econometrica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1038\/376046a0","article-title":"Scaling behaviour in the dynamics of an economic index","volume":"376","author":"Mantegna","year":"1995","journal-title":"Nature"},{"key":"ref_15","first-page":"509","article-title":"Truncated L\u00e9vy walks and an emerging market economic index","volume":"297","author":"Riera","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6519","DOI":"10.1103\/PhysRevE.60.6519","article-title":"Scaling of the distribution of price fluctuations of individual companies","volume":"60","author":"Plerou","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5305","DOI":"10.1103\/PhysRevE.60.5305","article-title":"Scaling of the distribution of fluctuations of financial market indices","volume":"60","author":"Gopikrishnan","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s100510050292","article-title":"Inverse cubic law for the distribution of stock price variations","volume":"3","author":"Gopikrishnan","year":"1998","journal-title":"Eur. Phys. J. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1080\/096031096333917","article-title":"The stable Paretian hypothesis and the frequency of large returns: An examination of major German stocks","volume":"6","author":"Lux","year":"1996","journal-title":"Appl. Financ. Econ."},{"key":"ref_20","first-page":"1487","article-title":"Fluctuations of WIG-the index of Warsaw Stock Exchange. Preliminary studies","volume":"32","author":"Makowiec","year":"2001","journal-title":"Acta Phys. Pol. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/S0378-4371(01)00289-8","article-title":"Quantifying the dynamics of financial correlations","volume":"299","author":"Ruf","year":"2001","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Takayasu, H. (2002). Variety of stock returns in normal and extreme market days: The August 1998 crisis. Empirical Science of Financial Fluctuations, Springer.","DOI":"10.1007\/978-4-431-66993-7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/S0378-4371(02)01216-5","article-title":"Dynamics of price and trading volume in a spin model of stock markets with heterogeneous agents","volume":"316","author":"Kaizoji","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_24","first-page":"4293","article-title":"Are the contemporary financial fluctuations sooner converging to normal?","volume":"34","author":"Ruf","year":"2003","journal-title":"Acta Phys. Pol. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1142\/S0218348X03002142","article-title":"Dynamical behavior of continuous tick data in futures exchange market","volume":"11","author":"Kim","year":"2003","journal-title":"Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1016\/j.physa.2004.05.052","article-title":"Herd behaviors in the stock and foreign exchange markets","volume":"341","author":"Kim","year":"2004","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/j.physa.2005.03.001","article-title":"On fitting the Pareto\u2013L\u00e9vy distribution to stock market index data: Selecting a suitable cutoff value","volume":"354","year":"2005","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Chatterjee, A., and Chakrabarti, B.K. (2006). The power (law) of Indian markets: Analysing NSE and BSE trading statistics. Econophysics of Stock and Other Markets. Proc. Econophys-Kolkata II, Springer.","DOI":"10.1007\/978-88-470-0502-0"},{"key":"ref_29","unstructured":"Oh, G., Kim, S., and Um, C.-J. (2006). Statistical properties of the returns of stock prices of international markets. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.physa.2006.07.035","article-title":"Nonextensive statistical features of the Polish stock market fluctuations","volume":"374","author":"Rak","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.physa.2007.05.007","article-title":"Statistical properties of daily ensemble variables in the Chinese stock markets","volume":"383","author":"Gu","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.physa.2007.04.130","article-title":"Stock market return distributions: From past to present","volume":"383","author":"Forczek","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"056109","DOI":"10.1103\/PhysRevE.79.056109","article-title":"Statistical analysis of the overnight and daytime return","volume":"79","author":"Wang","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physrep.2012.01.007","article-title":"Physical approach to complex systems","volume":"515","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"121055","DOI":"10.1016\/j.physa.2019.121055","article-title":"Fat tails in financial return distributions revisited: Evidence from the Korean stock market","volume":"526","author":"Eom","year":"2019","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1016\/j.eneco.2019.05.015","article-title":"Multifractal cross-correlations between the world oil and other financial markets","volume":"81","author":"Stanuszek","year":"2019","journal-title":"Energy Econ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"045103","DOI":"10.1103\/PhysRevE.66.045103","article-title":"Different scaling behaviors of commodity spot and future prices","volume":"66","author":"Matia","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"071101","DOI":"10.1063\/1.5036517","article-title":"Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects","volume":"28","author":"Minati","year":"2018","journal-title":"Chaos"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.10.005","article-title":"Multiscale characteristics of the emerging global cryptocurrency market","volume":"901","author":"Minati","year":"2021","journal-title":"Phys. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"012124","DOI":"10.1088\/1742-6596\/1730\/1\/012124","article-title":"Recent scaling properties of Bitcoin price returns","volume":"1730","author":"Takaishi","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_41","first-page":"291","article-title":"Comovements in stock prices in the very short run","volume":"74","author":"Epps","year":"1979","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"066126","DOI":"10.1103\/PhysRevE.65.066126","article-title":"Random matrix approach to cross correlations in financial data","volume":"65","author":"Plerou","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.physa.2003.09.012","article-title":"Alternation of different fluctuation regimes in the stock market dynamics","volume":"330","author":"Speth","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.physa.2004.01.050","article-title":"Time scales involved in emergent market coherence","volume":"337","author":"Speth","year":"2004","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_45","first-page":"3039","article-title":"Asymmetric matrices in an analysis of financial correlations","volume":"37","year":"2006","journal-title":"Acta Phys. Pol. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.2307\/2938368","article-title":"Long term memory in stock market prices","volume":"59","author":"Lo","year":"1991","journal-title":"Econometrica"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1080\/758534941","article-title":"Is there long-term memory in UK stock returns?","volume":"3","author":"Mills","year":"1993","journal-title":"Appl. Financ. Econ."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1086\/261535","article-title":"Permanent and temporary components in stock prices","volume":"96","author":"Fama","year":"1988","journal-title":"J. Polit. Econ."},{"key":"ref_49","first-page":"50","article-title":"Long memory in emerging market stock returns","volume":"5","author":"Wright","year":"2001","journal-title":"Emerg. Mark. Quart."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1080\/09603100010025733","article-title":"Long memory in stock returns: Some international evidence","volume":"12","author":"Henry","year":"2002","journal-title":"J. Appl. Financ. Econ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/j.physa.2005.09.051","article-title":"Fractionally integrated processes for transition economies","volume":"362","author":"Podobnik","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.ememar.2013.11.001","article-title":"Do emerging markets become more efficient as they develop? Long memory persistence in equity indices","volume":"18","author":"Hull","year":"2014","journal-title":"Emerg. Mark. Rev."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0927-5398(93)90006-D","article-title":"A long memory property of stock market returns and a new model","volume":"1","author":"Ding","year":"1993","journal-title":"J. Empir. Financ."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0304-4076(95)01732-1","article-title":"Long memory processes and fractional integration in econometrics","volume":"73","author":"Baillie","year":"1996","journal-title":"J. Econ."},{"key":"ref_55","first-page":"1","article-title":"The long memory of the efficient market","volume":"8","author":"Lillo","year":"2004","journal-title":"Stud. Nonlinear Dyn. Econom."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"383","DOI":"10.2307\/2325486","article-title":"Efficient capital markets: A review of theory and empirical work","volume":"25","author":"Fama","year":"1970","journal-title":"J. Financ."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"5643","DOI":"10.1016\/j.physa.2012.06.051","article-title":"Is the US stock market becoming weakly efficient over time? Evidence from 80-year-long data","volume":"391","author":"Rodriguez","year":"2012","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1038\/nature01624","article-title":"A theory of power-law distributions in financial market fluctuations","volume":"423","author":"Gabaix","year":"2003","journal-title":"Nature"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1088\/1469-7688\/4\/1\/C01","article-title":"On the origin of power law tails in price fluctuations","volume":"4","author":"Farmer","year":"2004","journal-title":"Quant. Financ."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1080\/14697680600835688","article-title":"There\u2019s more to volatility than volume","volume":"6","author":"Gillemot","year":"2006","journal-title":"Quant. Financ."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1080\/14697688.2017.1397283","article-title":"Linear models for the impact of order flow on prices II: The mixture transition distribution model","volume":"18","author":"Taranto","year":"2018","journal-title":"Quant. Financ."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1142\/S0219525903000906","article-title":"Exponential laws of stock price index and a stochastic model","volume":"6","author":"Kaizoji","year":"2003","journal-title":"Adv. Compl. Syst."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.physa.2004.06.122","article-title":"Exponential distribution of financial returns at mesoscopic time lags: A new stylized fact","volume":"344","author":"Silva","year":"2004","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1080\/14697680500151343","article-title":"Empirical distributions of stock returns: Between the stretched exponential and the power law?","volume":"5","author":"Malevergne","year":"2005","journal-title":"Quant. Financ."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/09603100500391008","article-title":"On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns","volume":"16","author":"Malevergne","year":"2006","journal-title":"Appl. Financ. Econ."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.physa.2006.10.099","article-title":"Non-extensive behavior of a stock market index at microscopic time scales","volume":"377","author":"Cortines","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"4787","DOI":"10.1016\/j.physa.2009.08.009","article-title":"Scaling and memory in the return intervals of realized volatility","volume":"388","author":"Ren","year":"2009","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.physa.2004.06.116","article-title":"Statistical properties of the Indonesian stock exchange index","volume":"344","author":"Mart","year":"2004","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.physa.2005.12.039","article-title":"Dynamics of the return distribution in the Korean financial market","volume":"363","author":"Yang","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_70","unstructured":"Scalas, E., and Kim, K. (2007). The art of fitting financial time series with L\u00e9vy stable distributions. arXiv."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1016\/j.physa.2012.11.026","article-title":"Scaling, stability and distribution of the high-frequency returns of the IBEX35 index","volume":"392","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.5506\/APhysPolB.44.2035","article-title":"Stock returns versus trading volume: Is the correspondence more general?","volume":"44","author":"Rak","year":"2013","journal-title":"Acta Phys. Pol. B"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.physa.2018.06.131","article-title":"Scaling properties of extreme price fluctuations in Bitcoin markets","volume":"510","author":"Stanley","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1257\/.41.2.478","article-title":"Forecasting Volatility in Financial Markets: A Review","volume":"41","author":"Poon","year":"2003","journal-title":"J. Econ. Lit."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Solomon, S. (1998). Stochastic Lotka-Volterra systems of competing auto-catalytic agents lead generically to truncated Pareto power wealth distribution, truncated Levy distribution of market returns, clustered volatility, booms and craches. Decision Technologies for Computational Finance, Springer.","DOI":"10.1007\/978-1-4615-5625-1_6"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/S0378-4371(01)00295-3","article-title":"Power laws of wealth, market order volumes and market returns","volume":"299","author":"Solomon","year":"2001","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1353\/mcb.2002.0004","article-title":"On rational bubbles and fat tails","volume":"34","author":"Lux","year":"2002","journal-title":"J. Money Credit Bank."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/S0378-4371(01)00281-3","article-title":"From rational bubbles to crashes","volume":"299","author":"Sornette","year":"2001","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0378-4371(02)00614-3","article-title":"\u201cSlimming\u201d of power law tails by increasing market returns","volume":"309","author":"Sornette","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1088\/1469-7688\/2\/6\/303","article-title":"Probability distribution of returns in the Heston model with stochastic volatility","volume":"2","author":"Yakovenko","year":"2002","journal-title":"Quant. Financ."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.physa.2005.08.037","article-title":"A theory of fluctuations in stock prices","volume":"363","author":"Bassler","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"P11013","DOI":"10.1088\/1742-5468\/2008\/11\/P11013","article-title":"Probability distribution of returns in the exponential Ornstein-Uhlenbeck model","volume":"2008","author":"Bormetti","year":"2008","journal-title":"J. Stat. Mech."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"065102(R)","DOI":"10.1103\/PhysRevE.80.065102","article-title":"Model for non-Gaussian intraday stock returns","volume":"80","author":"Gerig","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1038\/381767a0","article-title":"Turbulent cascades in foreign exchange markets","volume":"381","author":"Ghashghaie","year":"1996","journal-title":"Nature"},{"key":"ref_85","unstructured":"Mandelbrot, B.B., Fisher, A., and Calvet, L. (2021, May 26). Multifractal Model of Asset Returns. Cowles Foundation Discussion Paper no. 1164. Available online: https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=78588."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1142\/S021902490000019X","article-title":"A stochastic cascade model for FX dynamics","volume":"3","author":"Breymann","year":"1996","journal-title":"Int. J. Theor. Appl. Financ."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/S0378-4371(97)00401-9","article-title":"Price variations in a stock market with many agents","volume":"246","author":"Bak","year":"1997","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1209\/epl\/i1997-00491-5","article-title":"A prototype model of stock exchange","volume":"40","author":"Caldarelli","year":"1997","journal-title":"EPL"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1017\/S1365100500015029","article-title":"Herd behavior and aggregate fluctuations in financial markets","volume":"4","author":"Cont","year":"2000","journal-title":"Macroecon. Dyn."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s100510070240","article-title":"Self-organized model for information spread in financial markets","volume":"16","author":"Zhang","year":"2000","journal-title":"Eur. Phys. J. B"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/S0378-4371(01)00103-0","article-title":"Stylized facts of financial markets and market crashes in minority game","volume":"294","author":"Challet","year":"2001","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1142\/S0129183101001845","article-title":"Expectation bubbles in a spin model of markets: Intermittency from frustration across scales","volume":"12","author":"Bornholdt","year":"2001","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/s100510050276","article-title":"Stretched exponential distributions in nature and economy: \u201cFat tails\u201d with characteristic scales","volume":"2","author":"Sornette","year":"1998","journal-title":"Eur. Phys. J. B"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1209\/epl\/i2003-10267-y","article-title":"Scale-dependent price fluctuations for the Indian stock market","volume":"66","author":"Matia","year":"2004","journal-title":"Europhys. Lett."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/j.physa.2005.10.015","article-title":"New statistic for financial return distributions: Power-law or exponential?","volume":"366","author":"Pisarenko","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1002\/ijfe.149","article-title":"A model for stock return distribution","volume":"6","author":"Linden","year":"2001","journal-title":"Int. J. Financ. Econ."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of the Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_98","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1016\/S0378-4371(02)01558-3","article-title":"Financial market dynamics","volume":"320","author":"Michael","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"066103","DOI":"10.1103\/PhysRevE.82.066103","article-title":"Nonuniversal distributions of stock returns in an emerging market","volume":"82","author":"Mu","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"60003","DOI":"10.1209\/0295-5075\/88\/60003","article-title":"Quantitative features of multifractal subtleties in time series","volume":"88","author":"Rak","year":"2009","journal-title":"EPL"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.physa.2004.06.132","article-title":"On anomalous distributions in intra-day financial time series and non-extensive statistical mechanics","volume":"344","year":"2004","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1093\/rfs\/1.1.41","article-title":"Stock market prices do not follow random walks: Evidence from a simple specification test","volume":"1","author":"Lo","year":"1988","journal-title":"Rev. Financ. Stud."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"102","DOI":"10.3905\/jpm.24.2.102","article-title":"Distributional characteristics of emerging market returns and asset allocation","volume":"24","author":"Bekaert","year":"1998","journal-title":"J. Portf. Manag."},{"key":"ref_105","unstructured":"Lee, K.E., and Lee, J.W. (2004). Scaling properites of price changes for Korean stock indices. arXiv."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.physa.2005.09.044","article-title":"Scaling analysis on Indian foreign exchange market","volume":"364","author":"Sarkar","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.physa.2005.06.095","article-title":"Underlying dynamics of typical fluctuations of an emerging market price index: The Heston model from minutes to months","volume":"361","author":"Vicente","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"2990","DOI":"10.1016\/j.physa.2012.01.023","article-title":"On the scaling of the distribution of daily price fluctuations in Mexican financial market index","volume":"391","author":"Alfonso","year":"2012","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.5506\/APhysPolB.43.2021","article-title":"Cross-correlations Between WTI Crude Oil Market and U.S. Stock Market: A Perspective from Econophysicss","volume":"45","author":"Gang","year":"2012","journal-title":"Acta Phys. Pol. B"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1007\/s11071-013-0895-7","article-title":"Cross-correlations between the CSI 300 spot and futures markets","volume":"73","author":"Gang","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_111","first-page":"65","article-title":"The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments","volume":"25","author":"Samuelson","year":"1970","journal-title":"Rev. Econ. Stud."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"15","DOI":"10.2307\/2330926","article-title":"Skewness preference and portfolio choice","volume":"17","author":"Kane","year":"1977","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_113","first-page":"897","article-title":"Co-skewness and capital asset pricing","volume":"35","author":"Friend","year":"1980","journal-title":"J. Financ."},{"key":"ref_114","first-page":"147","article-title":"Models of stock returns. A comparison","volume":"39","author":"Kon","year":"1984","journal-title":"J. Financ."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1016\/0378-4266(90)90009-Q","article-title":"Statistical study of foreign exchange rates, empirical evidence of a price change scaling law and intraday analysis","volume":"14","author":"Dacorogna","year":"1990","journal-title":"J. Bank. Financ."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s007800050018","article-title":"From the bird\u2019s eye to the microscope: A survey of new stylized facts of the intra-day foreign exchange markets","volume":"1","author":"Guillaume","year":"1997","journal-title":"Financ. Stoch."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.physa.2007.04.122","article-title":"Assessing symmetry of financial returns series","volume":"383","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_118","doi-asserted-by":"crossref","unstructured":"Derksen, M., Kleijn, B., and de Vilder, R. (2020). Heavy tailed distributions in closing auctions. arXiv.","DOI":"10.2139\/ssrn.3751309"},{"key":"ref_119","doi-asserted-by":"crossref","unstructured":"Mi\u015bkiewicz, J. (2021). Network analysis of cross-correlations on Forex market during crises. Globalisation on Forex market. Entropy, 23.","DOI":"10.3390\/e23030352"},{"key":"ref_120","unstructured":"Nakamoto, S. (2021, May 26). Bitcoin: A Peer-to-Peer Electronic Cash System. Available online: https:\/\/git.dhimmel.com\/bitcoin-whitepaper\/."},{"key":"ref_121","unstructured":"(2021, May 26). Ethereum. Available online: http:\/\/www.ethereum.org."},{"key":"ref_122","unstructured":"(2021, May 26). Dukascopy. Available online: https:\/\/www.dukascopy.com."},{"key":"ref_123","unstructured":"(2021, May 26). Kraken. Available online: http:\/\/www.kraken.com."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"105003","DOI":"10.1088\/1367-2630\/12\/10\/105003","article-title":"The foreign exchange market: Return distributions, multifractality, anomalous multifractality and the Epps effect","volume":"12","author":"Rak","year":"2010","journal-title":"New J. Phys."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/S0378-4371(01)00119-4","article-title":"Towards identifying the world stock market cross-correlations: DAX versus Dow Jones","volume":"294","author":"Ruf","year":"2001","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_126","doi-asserted-by":"crossref","unstructured":"Takayasu, H. (2002). Speth, J. Dynamics of correlations in the stock market. Empirical Science of Financial Fluctuations, Springer.","DOI":"10.1007\/978-4-431-66993-7"},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"2349","DOI":"10.1007\/s11071-019-05335-5","article-title":"Detecting correlations and triangular arbitrage opportunities in the Forex by means of multifractal detrended cross-correlations analysis","volume":"98","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_128","doi-asserted-by":"crossref","unstructured":"Dro\u017cd\u017c, S., Minati, L., O\u015bwi\u0119cimka, P., Stanuszek, M., and W\u0105torek, M. (2019). Signatures of crypto-currency market decoupling from the Forex. Future Internet, 11.","DOI":"10.3390\/fi11070154"},{"key":"ref_129","doi-asserted-by":"crossref","unstructured":"Dro\u017cd\u017c, S., Kwapie\u0144, J., O\u015bwi\u0119cimka, P., Stanisz, T., and W\u0105torek, M. (2020). Complexity in economic and social systems: Cryptocurrency market at around COVID-19. Entropy, 22.","DOI":"10.3390\/e22091043"},{"key":"ref_130","doi-asserted-by":"crossref","unstructured":"Muvunza, T. (2020). An \u03b1-stable approach to modelling highly speculative assets and cryptocurrencies. arXiv.","DOI":"10.2139\/ssrn.3505859"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"023122","DOI":"10.1063\/1.5139634","article-title":"Competition of noise and collectivity in global cryptocurrency trading: Route to a self-contained market","volume":"30","author":"Minati","year":"2020","journal-title":"Chaos"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/884\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:29:14Z","timestamp":1760164154000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/884"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,12]]},"references-count":131,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["e23070884"],"URL":"https:\/\/doi.org\/10.3390\/e23070884","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,12]]}}}