Multifractal analysis of Bitcoin price dynamics
Abstract
This research employs Multifractal Detrended Fluctuation Analysis (MFDFA) to investigate multifractal properties in financial variables, including Bitcoin prices and economic indicators. Spanning 2019–2022, the analysis reveals multifractal scaling not only in Bitcoin prices, but also in economic indicators such as inflation rates and energy commodity prices. The non-linear singularity spectra unveil the multifaceted nature of scaling properties. Temporal analysis exposes intriguing trends in multifractality with implications for market efficiency. Furthermore, correlation analysis unveils connections among multifractal properties. For instance, a positive correlation between oil prices and Bitcoin suggests similar market forces. The log-log plot of fluctuation function Fq versus lag size demonstrates a power-law relationship, characteristic of multifractal systems. The empirical data’s alignment in log-log space suggests self-similarity in the Bitcoin time series, supporting multifractality. The calculated Hurst exponents values suggest varying degrees of multifractality across the years, with 2021 exhibiting the highest degree and 2022 the lowest. Furthermore, an asymmetry index (0.5767) deviating from 0.5 indicates that the multifractal nature of the Bitcoin market is not symmetric. This research enhances risk assessment and portfolio optimization in finance. It challenges the Efficient Market Hypothesis (EMH), emphasizing the significance of MFDFA in comprehending financial market and economic factor’s relationships.
Keyword : multifractal analysis, financial time series, Bitcoin, market efficiency, singularity spectra, correlation matrix
How to Cite
Bucur, C., Tudorică, B.-G., Bâra, A., & Oprea, S.-V. (2025). Multifractal analysis of Bitcoin price dynamics. Journal of Business Economics and Management, 26(1), 21–48. https://doi.org/10.3846/jbem.2025.23025
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Agnese, P. (2021). On blockchains, cryptos, and media of exchange. Not there (yet). International Journal of Intellectual Property Management, 11(1), 81–94. https://doi.org/10.1504/ijipm.2021.113359
Alaoui, M. El, Bouri, E., & Roubaud, D. (2019). Bitcoin price–volume: A multifractal cross-correlation approach. Finance Research Letters, 31, 374–381. https://doi.org/10.1016/j.frl.2018.12.011
Aslam, F., Ferreira, P., & Mohti, W. (2023a). Investigating efficiency of frontier stock markets using multifractal detrended fluctuation analysis. International Journal of Emerging Markets, 18(7), 1650–1676. https://doi.org/10.1108/IJOEM-11-2020-1348
Aslam, F., Zil-E-Huma, Bibi, R., & Ferreira, P. (2023b). The nexus between twitter-based uncertainty and cryptocurrencies: A multifractal analysis. Fractals, 31(3). https://doi.org/10.1142/S0218348X23500275
Aysan, A. F., Khan, A. U. I., Isac, N., & Drammeh, O. (2023). Threat of intervention in cryptocurrency market: West side story of Bitcoin and Ripple. Economic Computation and Economic Cybernetics Studies and Research, 57(4), 41–56. https://doi.org/10.24818/18423264/57.4.23.03
Bǎra, A., Georgescu, I. A., Oprea, S. V., & Cristescu, M. P. (2024). Exploring the dynamics of brent crude oil, S&P500 and Bitcoin prices Amid Economic Instability. IEEE Access, 12, 31366–31385. https://doi.org/10.1109/ACCESS.2024.3370029
Bielinskyi, A., Soloviev, V., Solovieva, V., Matviychuk, A., & Semerikov, S. (2023). The analysis of multifractal cross-correlation connectedness between Bitcoin and the stock market. Lecture Notes on Data Engineering and Communications Technologies, 178, 323–345. https://doi.org/10.1007/978-3-031-35467-0_21
Cao, G., & Ling, M. (2022). Asymmetry and conduction direction of the interdependent structure between cryptocurrency and US dollar, renminbi, and gold markets. Chaos, Solitons and Fractals, 155, Article 111671. https://doi.org/10.1016/j.chaos.2021.111671
Cao, G., & Xie, W. (2021). The impact of the shutdown policy on the asymmetric interdependence structure and risk transmission of cryptocurrency and China’s financial market. North American Journal of Economics and Finance, 58, Article 101514. https://doi.org/10.1016/j.najef.2021.101514
Cheung, A. (W.-K.), Roca, E., & Su, J. J. (2015). Crypto-currency bubbles: An application of the Phillips–Shi–Yu (2013) methodology on Mt. Gox bitcoin prices. Applied Economics, 47(23), 2348–2358. https://doi.org/10.1080/00036846.2015.1005827
Chowdhury, M. A. F., Abdullah, M., Alam, M., Abedin, M. Z., & Shi, B. (2023). NFTs, DeFi, and other assets efficiency and volatility dynamics: An asymmetric multifractality analysis. International Review of Financial Analysis, 87, Article 102642. https://doi.org/10.1016/j.irfa.2023.102642
da Silva Filho, A. C., Maganini, N. D., & de Almeida, E. F. (2018). Multifractal analysis of Bitcoin market. Physica A: Statistical Mechanics and its Applications, 512, 954–967. https://doi.org/10.1016/j.physa.2018.08.076
Fernandes, L. H. S., Kristjanpoller, W., & Tabak, B. M. (2023). Asymmetric multifractal cross-correlation dynamics between fiat currencies and cryptocurrencies. Fractals 31(1), Article 2350006. https://doi.org/10.1142/S0218348X23500068
Fernandes, L. H. S., Silva, J. W. L., & de Araujo, F. H. A. (2022). Multifractal risk measures by Macroeconophysics perspective: The case of Brazilian inflation dynamics. Chaos, Solitons and Fractals, 158, Article 112052. https://doi.org/10.1016/j.chaos.2022.112052
Fu, X. L., Gao, X. L., Shan, Z., Ma, Y. J., Jiang, Z. Q., & Zhou, W. X. (2023). Multifractal characteristics and return predictability in the Chinese stock markets. Annals of Operations Research. https://doi.org/10.1007/s10479-023-05281-x
Gajardo, G., Kristjanpoller, W. D., & Minutolo, M. (2018). Does Bitcoin exhibit the same asymmetric multifractal cross-correlations with crude oil, gold and DJIA as the Euro, Great British Pound and Yen? Chaos, Solitons and Fractals, 109, 195–205. https://doi.org/10.1016/j.chaos.2018.02.029
Ghosh, B., & Kozarevic, E. (2019). Multifractal analysis of volatility for detection of herding and bubble: Evidence from CNX Nifty HFT. In Investment Management and Financial Innovations, 16(3), 182–193. https://doi.org/10.21511/imfi.16(3).2019.17
Gunay, S., & Kaşkaloğlu, K. (2019). Seeking a chaotic order in the cryptocurrency market. Mathematical and Computational Applications, 24(2), 36. https://doi.org/10.3390/mca24020036
Huang, X., & Tang, H. (2022). Measuring multi-volatility states of financial markets based on multifractal clustering model. Journal of Forecasting, 41(3), 422–434. https://doi.org/10.1002/for.2820
Humayun, M., & Belk, R. W. (2018). “Satoshi is dead. Long live satoshi”: The curious case of bitcoin’s creator. In S. N. N. Cross, C. Ruvalcaba, A. Venkatesh, & R. W. Belk (Eds.), Research in consumer behavior (vol. 19, pp. 19–35). Emerald Publishing Limited. https://doi.org/10.1108/S0885-211120180000019002
Jiang, C., Dev, P., & Maller, R. A. (2020). A hypothesis test method for detecting multifractal scaling, applied to Bitcoin prices. Journal of Risk and Financial Management, 13(5), Article 104. https://doi.org/10.3390/jrfm13050104
Jiang, Z. Q., Xie, W. J., Zhou, W. X., & Sornette, D. (2019). Multifractal analysis of financial markets: A review. Reports on Progress in Physics, 82(12), Article 125901. https://doi.org/10.1088/1361-6633/ab42fb
Jin, J., Yu, J., Hu, Y., & Shang, Y. (2019). Which one is more informative in determining price movements of hedging assets? Evidence from Bitcoin, gold and crude oil markets. Physica A: Statistical Mechanics and Its Applications, 527, Article 121121. https://doi.org/10.1016/j.physa.2019.121121
John, K., O’Hara, M., & Saleh, F. (2022). Bitcoin and beyond. Annual Review of Financial Economics, 14, 95–115. https://doi.org/10.1146/annurev-financial-111620-011240
Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H. E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications, 316(1–4), 87–114. https://doi.org/10.1016/S0378-4371(02)01383-3
Ke, J., Duan, Y., Xu, C., & Zhang, Y. (2023). Cross-correlation multifractal analysis of technological innovation, financial market and real economy indices. Fractal and Fractional, 7(3), Article 267. https://doi.org/10.3390/fractalfract7030267
Khan, K., Su, C. W., Khurshid, A., & Umar, M. (2022). COVID-19 impact on multifractality of energy prices: Asymmetric multifractality analysis. Energy, 256, Article 124607. https://doi.org/10.1016/j.energy.2022.124607
Khan, T., & Suresh, G. (2022). Do all shocks produce embedded herding and bubble? An empirical observation of the Indian stock market. Investment Management and Financial Innovations, 19(3), 346–359. https://doi.org/10.21511/imfi.19(3).2022.29
Khurshid, A., Khan, K., Cifuentes-Faura, J., & Chen, Y. (2024). Asymmetric multifractality: Comparative efficiency analysis of global technological and renewable energy prices using MFDFA and A-MFDFA approaches. Energy, 289, Article 130106. https://doi.org/10.1016/j.energy.2023.130106
Kristjanpoller, W., & Bouri, E. (2019). Asymmetric multifractal cross-correlations between the main world currencies and the main cryptocurrencies. Physica A: Statistical Mechanics and its Applications, 523, 1057–1071. https://doi.org/10.1016/j.physa.2019.04.115
Leonarduzzi, R., Abry, P., Wendt, H., Jaffard, S., & Touchette, H. (2019). A generalized multifractal formalism for the estimation of nonconcave multifractal spectra. IEEE Transactions on Signal Processing, 67(10), 110–119. https://doi.org/10.1109/TSP.2018.2879617
Li, Y., Yin, M., Khan, K., & Su, C. W. (2023). The impact of COVID-19 on shipping freights: Asymmetric multifractality analysis. Maritime Policy and Management, 50(7), 889–907. https://doi.org/10.1080/03088839.2022.2081372
Lyócsa, Š., Molnár, P., Plíhal, T., & Širaňová, M. (2020). Impact of macroeconomic news, regulation and hacking exchange markets on the volatility of bitcoin. Journal of Economic Dynamics and Control, 119, Article 103980. https://doi.org/10.1016/j.jedc.2020.103980
Ma, J., Wang, T., & Zhao, R. (2022). Quantifying cross-correlations between economic policy uncertainty and bitcoin market: Evidence from multifractal analysis. Discrete Dynamics in Nature and Society, 2022(1), Article 1072836. https://doi.org/10.1155/2022/1072836
Meng, K., & Khan, K. (2024). Is cryptocurrency efficient? A high-frequency asymmetric multifractality analysis. Computational Economics. https://doi.org/10.1007/s10614-023-10402-6
Mensah, I. K., & Mwakapesa, D. S. (2022). The drivers of the behavioral adoption intention of BITCOIN payment from the perspective of Chinese citizens. Security and Communication Networks, 2022, Article 7373658 (Retraction published 2024, Security and Communication Networks, 2024, Article 9827698). https://doi.org/10.1155/2022/7373658
Mensi, W., Lee, Y. J., Al-Yahyaee, K. H., Sensoy, A., & Yoon, S. M. (2019). Intraday downward/upward multifractality and long memory in Bitcoin and Ethereum markets: An asymmetric multifractal detrended fluctuation analysis. Finance Research Letters, 31, 19–25. https://doi.org/10.1016/j.frl.2019.03.029
Nguyen, K. Q. (2022). The correlation between the stock market and Bitcoin during COVID-19 and other uncertainty periods. Finance Research Letters, 46, Article 102284. https://doi.org/10.1016/j.frl.2021.102284
Oprea, S. V., Georgescu, I. A., & Bâra, A. (2024). Is Bitcoin ready to be a widespread payment method? Using price volatility and setting strategies for merchants. Electronic Commerce Research, 24, 1267–1305. https://doi.org/10.1007/s10660-024-09812-x
Rufino, C. C. (2023). On the volatility and market inefficiency of Bitcoin during the COVID-19 pandemic. DLSU Business and Economics Review, 32(2), 23–32.
Rydin Gorjão, L., Hassan, G., Kurths, J., & Witthaut, D. (2022). MFDFA: Efficient multifractal detrended fluctuation analysis in python. Computer Physics Communications, 273, Article 108254. https://doi.org/10.1016/j.cpc.2021.108254
Salat, H., Murcio, R., & Arcaute, E. (2017). Multifractal methodology. Physica A: Statistical Mechanics and its Applications, 473, 467–487. https://doi.org/10.1016/j.physa.2017.01.041
Segnon, M., & Bekiros, S. (2020). Forecasting volatility in bitcoin market. Annals of Finance, 16, 435–462. https://doi.org/10.1007/s10436-020-00368-y
Segnon, M., & Trede, M. (2018). Forecasting market risk of portfolios: copula-Markov switching multifractal approach. European Journal of Finance, 24(14), 1123–1143. https://doi.org/10.1080/1351847X.2017.1400453
Selmi, R., Mensi, W., Hammoudeh, S., & Bouoiyour, J. (2018). Is Bitcoin a hedge, a safe haven or a diversifier for oil price movements? A comparison with gold. Energy Economics, 74, Article 787–801. https://doi.org/10.1016/j.eneco.2018.07.007
Shen, N., & Chen, J. (2023). Asymmetric multifractal spectrum distribution based on detrending moving average cross-correlation analysis. Physica A: Statistical Mechanics and its Applications, 615, Article 128559. https://doi.org/10.1016/j.physa.2023.128559
Shrestha, K. (2021). Multifractal Detrended Fluctuation Analysis of Return on Bitcoin. International Review of Finance, 21(1), 312–323. https://doi.org/10.1111/irfi.12256
Stavroyiannis, S., Babalos, V., Bekiros, S., Lahmiri, S., & Uddin, G. S. (2019). The high frequency multifractal properties of Bitcoin. Physica A: Statistical Mechanics and its Applications, 520, 62–71. https://doi.org/10.1016/j.physa.2018.12.037
Stosic, D., Stosic, D., Ludermir, T. B., & Stosic, T. (2019). Multifractal behavior of price and volume changes in the cryptocurrency market. Physica A: Statistical Mechanics and its Applications, 520, 54–61. https://doi.org/10.1016/j.physa.2018.12.038
Sun, Y., Amanda, C., & Centana, B. C. (2023). The effect of hashrate, transaction volume, social media and macroeconomics on Bitcoin before and during the COVID-19 pandemic. Asian Journal of Accounting Research, 8(3), 293–306. https://doi.org/10.1108/AJAR-10-2022-0319
Takaishi, T. (2018). Statistical properties and multifractality of Bitcoin. Physica A: Statistical Mechanics and its Applications, 506, 507–519. https://doi.org/10.1016/j.physa.2018.04.046
Takaishi, T. (2020). Rough volatility of Bitcoin. Finance Research Letters, 32, Article 101379. https://doi.org/10.1016/j.frl.2019.101379
Takaishi, T. (2021). Time-varying properties of asymmetric volatility and multifractality in Bitcoin. PLoS ONE, 16(2), Article e0246209. https://doi.org/10.1371/journal.pone.0246209
Takaishi, T., & Adachi, T. (2020). Market efficiency, liquidity, and multifractality of Bitcoin: A dynamic study. Asia-Pacific Financial Markets, 27, 145–154. https://doi.org/10.1007/s10690-019-09286-0
Telli, Ş., & Chen, H. (2020). Multifractal behavior in return and volatility series of Bitcoin and gold in comparison. Chaos, Solitons and Fractals, 139, Article 109994. https://doi.org/10.1016/j.chaos.2020.109994
Telli, Ş., & Chen, H. (2021). Multifractal behavior relationship between crypto markets and Wikipedia-Reddit online platforms. Chaos, Solitons and Fractals, 152, Article 111331. https://doi.org/10.1016/j.chaos.2021.111331
Vaz, C., Pascoal, R., & Sebastião, H. (2021). Price appreciation and roughness duality in bitcoin: A multifractal analysis. Mathematics, 9(17), Article 2088. https://doi.org/10.3390/math9172088
Wang, Y., Sun, Q., Zhang, Z., & Chen, L. (2022). A risk measure of the stock market that is based on multifractality. Physica A: Statistical Mechanics and its Applications, 596, Article 127203. https://doi.org/10.1016/j.physa.2022.127203
Wątorek, M., Kwapień, J., & Drożdż, S. (2022). Multifractal cross-correlations of Bitcoin and ether trading characteristics in the post-COVID-19 time. Future Internet, 14(7), Article 215. https://doi.org/10.3390/fi14070215
Wenhao, X., & Guangxi, C. (2022). Research on multifractal characteristics of Bitcoin price-volume relationship based on MF-ADCCA method. Journal of Industrial Engineering and Engineering Management. https://doi.org/10.13587/j.cnki.jieem.2022.05.001
Yin, T., & Wang, Y. (2022). Nonlinear analysis and prediction of Bitcoin return’s volatility. E a M: Ekonomie a Management 25(2), 102–117. https://doi.org/10.15240/tul/001/2022-2-007
Zhang, W., Wang, P., Li, X., & Shen, D. (2018). Quantifying the cross-correlations between online searches and Bitcoin market. Physica A: Statistical Mechanics and its Applications, 509, 657–672. https://doi.org/10.1016/j.physa.2018.06.073
Zhang, X., Yang, L., & Zhu, Y. (2019). Analysis of multifractal characterization of Bitcoin market based on multifractal detrended fluctuation analysis. Physica A: Statistical Mechanics and its Applications, 523, 973–983. https://doi.org/10.1016/j.physa.2019.04.149
Alaoui, M. El, Bouri, E., & Roubaud, D. (2019). Bitcoin price–volume: A multifractal cross-correlation approach. Finance Research Letters, 31, 374–381. https://doi.org/10.1016/j.frl.2018.12.011
Aslam, F., Ferreira, P., & Mohti, W. (2023a). Investigating efficiency of frontier stock markets using multifractal detrended fluctuation analysis. International Journal of Emerging Markets, 18(7), 1650–1676. https://doi.org/10.1108/IJOEM-11-2020-1348
Aslam, F., Zil-E-Huma, Bibi, R., & Ferreira, P. (2023b). The nexus between twitter-based uncertainty and cryptocurrencies: A multifractal analysis. Fractals, 31(3). https://doi.org/10.1142/S0218348X23500275
Aysan, A. F., Khan, A. U. I., Isac, N., & Drammeh, O. (2023). Threat of intervention in cryptocurrency market: West side story of Bitcoin and Ripple. Economic Computation and Economic Cybernetics Studies and Research, 57(4), 41–56. https://doi.org/10.24818/18423264/57.4.23.03
Bǎra, A., Georgescu, I. A., Oprea, S. V., & Cristescu, M. P. (2024). Exploring the dynamics of brent crude oil, S&P500 and Bitcoin prices Amid Economic Instability. IEEE Access, 12, 31366–31385. https://doi.org/10.1109/ACCESS.2024.3370029
Bielinskyi, A., Soloviev, V., Solovieva, V., Matviychuk, A., & Semerikov, S. (2023). The analysis of multifractal cross-correlation connectedness between Bitcoin and the stock market. Lecture Notes on Data Engineering and Communications Technologies, 178, 323–345. https://doi.org/10.1007/978-3-031-35467-0_21
Cao, G., & Ling, M. (2022). Asymmetry and conduction direction of the interdependent structure between cryptocurrency and US dollar, renminbi, and gold markets. Chaos, Solitons and Fractals, 155, Article 111671. https://doi.org/10.1016/j.chaos.2021.111671
Cao, G., & Xie, W. (2021). The impact of the shutdown policy on the asymmetric interdependence structure and risk transmission of cryptocurrency and China’s financial market. North American Journal of Economics and Finance, 58, Article 101514. https://doi.org/10.1016/j.najef.2021.101514
Cheung, A. (W.-K.), Roca, E., & Su, J. J. (2015). Crypto-currency bubbles: An application of the Phillips–Shi–Yu (2013) methodology on Mt. Gox bitcoin prices. Applied Economics, 47(23), 2348–2358. https://doi.org/10.1080/00036846.2015.1005827
Chowdhury, M. A. F., Abdullah, M., Alam, M., Abedin, M. Z., & Shi, B. (2023). NFTs, DeFi, and other assets efficiency and volatility dynamics: An asymmetric multifractality analysis. International Review of Financial Analysis, 87, Article 102642. https://doi.org/10.1016/j.irfa.2023.102642
da Silva Filho, A. C., Maganini, N. D., & de Almeida, E. F. (2018). Multifractal analysis of Bitcoin market. Physica A: Statistical Mechanics and its Applications, 512, 954–967. https://doi.org/10.1016/j.physa.2018.08.076
Fernandes, L. H. S., Kristjanpoller, W., & Tabak, B. M. (2023). Asymmetric multifractal cross-correlation dynamics between fiat currencies and cryptocurrencies. Fractals 31(1), Article 2350006. https://doi.org/10.1142/S0218348X23500068
Fernandes, L. H. S., Silva, J. W. L., & de Araujo, F. H. A. (2022). Multifractal risk measures by Macroeconophysics perspective: The case of Brazilian inflation dynamics. Chaos, Solitons and Fractals, 158, Article 112052. https://doi.org/10.1016/j.chaos.2022.112052
Fu, X. L., Gao, X. L., Shan, Z., Ma, Y. J., Jiang, Z. Q., & Zhou, W. X. (2023). Multifractal characteristics and return predictability in the Chinese stock markets. Annals of Operations Research. https://doi.org/10.1007/s10479-023-05281-x
Gajardo, G., Kristjanpoller, W. D., & Minutolo, M. (2018). Does Bitcoin exhibit the same asymmetric multifractal cross-correlations with crude oil, gold and DJIA as the Euro, Great British Pound and Yen? Chaos, Solitons and Fractals, 109, 195–205. https://doi.org/10.1016/j.chaos.2018.02.029
Ghosh, B., & Kozarevic, E. (2019). Multifractal analysis of volatility for detection of herding and bubble: Evidence from CNX Nifty HFT. In Investment Management and Financial Innovations, 16(3), 182–193. https://doi.org/10.21511/imfi.16(3).2019.17
Gunay, S., & Kaşkaloğlu, K. (2019). Seeking a chaotic order in the cryptocurrency market. Mathematical and Computational Applications, 24(2), 36. https://doi.org/10.3390/mca24020036
Huang, X., & Tang, H. (2022). Measuring multi-volatility states of financial markets based on multifractal clustering model. Journal of Forecasting, 41(3), 422–434. https://doi.org/10.1002/for.2820
Humayun, M., & Belk, R. W. (2018). “Satoshi is dead. Long live satoshi”: The curious case of bitcoin’s creator. In S. N. N. Cross, C. Ruvalcaba, A. Venkatesh, & R. W. Belk (Eds.), Research in consumer behavior (vol. 19, pp. 19–35). Emerald Publishing Limited. https://doi.org/10.1108/S0885-211120180000019002
Jiang, C., Dev, P., & Maller, R. A. (2020). A hypothesis test method for detecting multifractal scaling, applied to Bitcoin prices. Journal of Risk and Financial Management, 13(5), Article 104. https://doi.org/10.3390/jrfm13050104
Jiang, Z. Q., Xie, W. J., Zhou, W. X., & Sornette, D. (2019). Multifractal analysis of financial markets: A review. Reports on Progress in Physics, 82(12), Article 125901. https://doi.org/10.1088/1361-6633/ab42fb
Jin, J., Yu, J., Hu, Y., & Shang, Y. (2019). Which one is more informative in determining price movements of hedging assets? Evidence from Bitcoin, gold and crude oil markets. Physica A: Statistical Mechanics and Its Applications, 527, Article 121121. https://doi.org/10.1016/j.physa.2019.121121
John, K., O’Hara, M., & Saleh, F. (2022). Bitcoin and beyond. Annual Review of Financial Economics, 14, 95–115. https://doi.org/10.1146/annurev-financial-111620-011240
Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H. E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications, 316(1–4), 87–114. https://doi.org/10.1016/S0378-4371(02)01383-3
Ke, J., Duan, Y., Xu, C., & Zhang, Y. (2023). Cross-correlation multifractal analysis of technological innovation, financial market and real economy indices. Fractal and Fractional, 7(3), Article 267. https://doi.org/10.3390/fractalfract7030267
Khan, K., Su, C. W., Khurshid, A., & Umar, M. (2022). COVID-19 impact on multifractality of energy prices: Asymmetric multifractality analysis. Energy, 256, Article 124607. https://doi.org/10.1016/j.energy.2022.124607
Khan, T., & Suresh, G. (2022). Do all shocks produce embedded herding and bubble? An empirical observation of the Indian stock market. Investment Management and Financial Innovations, 19(3), 346–359. https://doi.org/10.21511/imfi.19(3).2022.29
Khurshid, A., Khan, K., Cifuentes-Faura, J., & Chen, Y. (2024). Asymmetric multifractality: Comparative efficiency analysis of global technological and renewable energy prices using MFDFA and A-MFDFA approaches. Energy, 289, Article 130106. https://doi.org/10.1016/j.energy.2023.130106
Kristjanpoller, W., & Bouri, E. (2019). Asymmetric multifractal cross-correlations between the main world currencies and the main cryptocurrencies. Physica A: Statistical Mechanics and its Applications, 523, 1057–1071. https://doi.org/10.1016/j.physa.2019.04.115
Leonarduzzi, R., Abry, P., Wendt, H., Jaffard, S., & Touchette, H. (2019). A generalized multifractal formalism for the estimation of nonconcave multifractal spectra. IEEE Transactions on Signal Processing, 67(10), 110–119. https://doi.org/10.1109/TSP.2018.2879617
Li, Y., Yin, M., Khan, K., & Su, C. W. (2023). The impact of COVID-19 on shipping freights: Asymmetric multifractality analysis. Maritime Policy and Management, 50(7), 889–907. https://doi.org/10.1080/03088839.2022.2081372
Lyócsa, Š., Molnár, P., Plíhal, T., & Širaňová, M. (2020). Impact of macroeconomic news, regulation and hacking exchange markets on the volatility of bitcoin. Journal of Economic Dynamics and Control, 119, Article 103980. https://doi.org/10.1016/j.jedc.2020.103980
Ma, J., Wang, T., & Zhao, R. (2022). Quantifying cross-correlations between economic policy uncertainty and bitcoin market: Evidence from multifractal analysis. Discrete Dynamics in Nature and Society, 2022(1), Article 1072836. https://doi.org/10.1155/2022/1072836
Meng, K., & Khan, K. (2024). Is cryptocurrency efficient? A high-frequency asymmetric multifractality analysis. Computational Economics. https://doi.org/10.1007/s10614-023-10402-6
Mensah, I. K., & Mwakapesa, D. S. (2022). The drivers of the behavioral adoption intention of BITCOIN payment from the perspective of Chinese citizens. Security and Communication Networks, 2022, Article 7373658 (Retraction published 2024, Security and Communication Networks, 2024, Article 9827698). https://doi.org/10.1155/2022/7373658
Mensi, W., Lee, Y. J., Al-Yahyaee, K. H., Sensoy, A., & Yoon, S. M. (2019). Intraday downward/upward multifractality and long memory in Bitcoin and Ethereum markets: An asymmetric multifractal detrended fluctuation analysis. Finance Research Letters, 31, 19–25. https://doi.org/10.1016/j.frl.2019.03.029
Nguyen, K. Q. (2022). The correlation between the stock market and Bitcoin during COVID-19 and other uncertainty periods. Finance Research Letters, 46, Article 102284. https://doi.org/10.1016/j.frl.2021.102284
Oprea, S. V., Georgescu, I. A., & Bâra, A. (2024). Is Bitcoin ready to be a widespread payment method? Using price volatility and setting strategies for merchants. Electronic Commerce Research, 24, 1267–1305. https://doi.org/10.1007/s10660-024-09812-x
Rufino, C. C. (2023). On the volatility and market inefficiency of Bitcoin during the COVID-19 pandemic. DLSU Business and Economics Review, 32(2), 23–32.
Rydin Gorjão, L., Hassan, G., Kurths, J., & Witthaut, D. (2022). MFDFA: Efficient multifractal detrended fluctuation analysis in python. Computer Physics Communications, 273, Article 108254. https://doi.org/10.1016/j.cpc.2021.108254
Salat, H., Murcio, R., & Arcaute, E. (2017). Multifractal methodology. Physica A: Statistical Mechanics and its Applications, 473, 467–487. https://doi.org/10.1016/j.physa.2017.01.041
Segnon, M., & Bekiros, S. (2020). Forecasting volatility in bitcoin market. Annals of Finance, 16, 435–462. https://doi.org/10.1007/s10436-020-00368-y
Segnon, M., & Trede, M. (2018). Forecasting market risk of portfolios: copula-Markov switching multifractal approach. European Journal of Finance, 24(14), 1123–1143. https://doi.org/10.1080/1351847X.2017.1400453
Selmi, R., Mensi, W., Hammoudeh, S., & Bouoiyour, J. (2018). Is Bitcoin a hedge, a safe haven or a diversifier for oil price movements? A comparison with gold. Energy Economics, 74, Article 787–801. https://doi.org/10.1016/j.eneco.2018.07.007
Shen, N., & Chen, J. (2023). Asymmetric multifractal spectrum distribution based on detrending moving average cross-correlation analysis. Physica A: Statistical Mechanics and its Applications, 615, Article 128559. https://doi.org/10.1016/j.physa.2023.128559
Shrestha, K. (2021). Multifractal Detrended Fluctuation Analysis of Return on Bitcoin. International Review of Finance, 21(1), 312–323. https://doi.org/10.1111/irfi.12256
Stavroyiannis, S., Babalos, V., Bekiros, S., Lahmiri, S., & Uddin, G. S. (2019). The high frequency multifractal properties of Bitcoin. Physica A: Statistical Mechanics and its Applications, 520, 62–71. https://doi.org/10.1016/j.physa.2018.12.037
Stosic, D., Stosic, D., Ludermir, T. B., & Stosic, T. (2019). Multifractal behavior of price and volume changes in the cryptocurrency market. Physica A: Statistical Mechanics and its Applications, 520, 54–61. https://doi.org/10.1016/j.physa.2018.12.038
Sun, Y., Amanda, C., & Centana, B. C. (2023). The effect of hashrate, transaction volume, social media and macroeconomics on Bitcoin before and during the COVID-19 pandemic. Asian Journal of Accounting Research, 8(3), 293–306. https://doi.org/10.1108/AJAR-10-2022-0319
Takaishi, T. (2018). Statistical properties and multifractality of Bitcoin. Physica A: Statistical Mechanics and its Applications, 506, 507–519. https://doi.org/10.1016/j.physa.2018.04.046
Takaishi, T. (2020). Rough volatility of Bitcoin. Finance Research Letters, 32, Article 101379. https://doi.org/10.1016/j.frl.2019.101379
Takaishi, T. (2021). Time-varying properties of asymmetric volatility and multifractality in Bitcoin. PLoS ONE, 16(2), Article e0246209. https://doi.org/10.1371/journal.pone.0246209
Takaishi, T., & Adachi, T. (2020). Market efficiency, liquidity, and multifractality of Bitcoin: A dynamic study. Asia-Pacific Financial Markets, 27, 145–154. https://doi.org/10.1007/s10690-019-09286-0
Telli, Ş., & Chen, H. (2020). Multifractal behavior in return and volatility series of Bitcoin and gold in comparison. Chaos, Solitons and Fractals, 139, Article 109994. https://doi.org/10.1016/j.chaos.2020.109994
Telli, Ş., & Chen, H. (2021). Multifractal behavior relationship between crypto markets and Wikipedia-Reddit online platforms. Chaos, Solitons and Fractals, 152, Article 111331. https://doi.org/10.1016/j.chaos.2021.111331
Vaz, C., Pascoal, R., & Sebastião, H. (2021). Price appreciation and roughness duality in bitcoin: A multifractal analysis. Mathematics, 9(17), Article 2088. https://doi.org/10.3390/math9172088
Wang, Y., Sun, Q., Zhang, Z., & Chen, L. (2022). A risk measure of the stock market that is based on multifractality. Physica A: Statistical Mechanics and its Applications, 596, Article 127203. https://doi.org/10.1016/j.physa.2022.127203
Wątorek, M., Kwapień, J., & Drożdż, S. (2022). Multifractal cross-correlations of Bitcoin and ether trading characteristics in the post-COVID-19 time. Future Internet, 14(7), Article 215. https://doi.org/10.3390/fi14070215
Wenhao, X., & Guangxi, C. (2022). Research on multifractal characteristics of Bitcoin price-volume relationship based on MF-ADCCA method. Journal of Industrial Engineering and Engineering Management. https://doi.org/10.13587/j.cnki.jieem.2022.05.001
Yin, T., & Wang, Y. (2022). Nonlinear analysis and prediction of Bitcoin return’s volatility. E a M: Ekonomie a Management 25(2), 102–117. https://doi.org/10.15240/tul/001/2022-2-007
Zhang, W., Wang, P., Li, X., & Shen, D. (2018). Quantifying the cross-correlations between online searches and Bitcoin market. Physica A: Statistical Mechanics and its Applications, 509, 657–672. https://doi.org/10.1016/j.physa.2018.06.073
Zhang, X., Yang, L., & Zhu, Y. (2019). Analysis of multifractal characterization of Bitcoin market based on multifractal detrended fluctuation analysis. Physica A: Statistical Mechanics and its Applications, 523, 973–983. https://doi.org/10.1016/j.physa.2019.04.149