Tail and memory behaviour of the cryptocurrency prices and stock market indices

• Autorzy: Kończal Julia , Wronka Michał

Identyfikatory

DOI

10.14708/ma.v52i1.7305

Warianty tytułu

PL

Analiza ogonów rozkładu prawdopodobieństwa i pamięci kryptowalut oraz indeksów giełdowych

• Języki publikacji: EN

Abstrakty

EN

Cryptocurrency markets are characterised by high volatility, high returns and immaturity relative to equity markets. In this paper, we aim to gain insight into the statistical differences between the markets. To this end, we analyse Bitcoin (BTC) and Ethereum (ETH) prices, Dow Jones Industrial Average (DJIA) index and Hang Seng Index (HSI). We concentrate here on the comparison of two important characteristics of the data, namely the distribution and memory. We find that the normal inverse distribution can stand for a universal distribution, double Weibull is very well fitted to BTC and ETH, and Student's t to BTC and HSI. Memory analysis performed with the use of various statistics indicates that the log-returns of DJIA show the strongest dependence (but not the long memory). Moreover, analysis of the heteroskedatic effect shows that the implied volatility of HSI can be very well described by the generalised autoregressive conditional heteroskedasticy (GARCH) model. We note that the procedure for testing the tail and memory behaviour introduced here can be applied to other datasets.

PL

Rynki kryptowalut cechują się dużą zmiennością, wysokimi stopami zwrotu oraz mniejszym stopniem rozwoju infrastrukturalnego i instytucjonalnego w porównaniu do tradycyjnych rynków kapitałowych. W tym artykule, naszym celem jest ujęcie statystycznych różnic między tymi rynkami. Aby to osiągnąć, analizujemy logarytmiczne stopy zwrotu Bitcoina (BTC), Ethereum (ETH), indeksu Dow Jones Industrial Average (DJIA) oraz indeksu Hang Seng (HSI). Koncentrujemy się na porównaniu dwóch istotnych cech danych, mianowicie rozkładu prawdopodobieństwa oraz pamięci. Uważamy, że odwrócony rozkład normalny może stanowić uniwersalny rozkład, opisujący badane dane. Podwójny rozkład Weibulla jest bardzo dobrze dopasowany do logarytmicznych stóp zwrotu BTC i ETH, a rozkład t-Studenta do BTC i HSI. Analiza pamięci przeprowadzona z wykorzystaniem różnych narzędzi statystycznych sugeruje, że logarytmiczne stopy zwrotu indeksu DJIA cechują się najsilniejszą zależnością (lecz nie wykazują cechy długiej pamięci). Dodatkowo, analiza heterdoskedastyczności logarytmicznych stóp zwrotu pokazuje, że implikowana zmienność logarytmicznych stóp zwrotu indeksu HSI przy pieniądzu (ang. At-The-Money) może być bardzo dobrze zreplikowana za pomocą filtracji modelu GARCH. Zauważamy, że metody analizy wykorzystane do badania ogonów oraz pamięci mogą zostać wykorzystane dla innych klas aktywów.

Słowa kluczowe

EN

heavy-tailed distribution; long memory; heteroskedasticity; stock index; cryptocurrency

PL

rozkłady ciężkoogonowe; długa pamięć; heteroskedastyczność; indeks giełdowy; kryptowaluty

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2024
• Tom: Vol. 52, no. 1
• Strony: 85--117
• Opis fizyczny: Bibliogr. 40 poz., tab., wykr.

Twórcy

autor

Kończal Julia

• Wrocław University of Science and Technology Faculty of Pure and Applied Mathematics Wybrzeże Wyspiańskiego 27, 50-370 Wrocław

• https://orcid.org/0009-0009-7237-948X

autor

Wronka Michał

• Wrocław University of Science and Technology Faculty of Pure and Applied Mathematics Wybrzeże Wyspiańskiego 27, 50-370 Wrocław

• https://orcid.org/0009-0003-1502-8735

Bibliografia

• [1] N. Ahmad and S. Rusgianto. Investigating the presence of long memory in DJIM index yield spreads. Procedia Economics and Finance, 7:73-79, 2013. Cited on p. 86.
• [2] N. Balakrishnan and S. Kocherlakota. On the double Weibull distribution: order statistics and estimation. Sankhyā: The Indian Journal of Statistics, Series B, pages 161-178, 1985. Cited on p. 91.
• [3] A. F. Bariviera, M. J. Basgall, W. Hasperué, and M. Naiouf. Some stylized facts of the Bitcoin market. Physica A: Statistical Mechanics and its Applications, 484:82-90, 2017. Cited on p. 86.
• [4] D. G. Baur, K. Hong, and A. D. Lee. Bitcoin: Medium of exchange or speculative assets? Journal of International Financial Markets, Institutions and Money, 54:177-189, 2018. Cited on p. 86.
• [5] J. Beran, Y. Feng, S. Ghosh, and R. Kulik. Long-Memory Processes: Probabilistic Properties and Statistical Methods. Springer, Berlin, 2013. Cited on pp. 97 and 98.
• [6] P. Bloomfield. Fourier Analysis of Time Series: An Introduction. John Wiley & Sons, 2004. Cited on p. 96.
• [7] T. Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3):307-327, 1986. Cited on p. 101.
• [8] G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung. Time Series Analysis: Forecasting and Control. John Wiley & Sons, 2015. Cited on p. 95.
• [9] P. J. Brockwell and R. A. Davis. Time Series: Theory and Methods. Springer, New York, 2009. Cited on p. 96.
• [10] K. Burnecki, J. Gajda, and G. Sikora. Stability and lack of memory of the returns of the Hang Seng Index. Physica A: Statistical Mechanics and its Applications, 390:3136-3146, 2011. Cited on pp. 86 and 99.
• [11] K. Burnecki and A. Weron. Fractional lévy stable motion can model subdiffusive dynamics. Phys. Rev. E, 82:021130, Aug 2010. Cited on p. 99.
• [12] K. Burnecki and A. Weron. Algorithms for testing of fractional dynamics: a practical guide to ARFIMA modelling. Journal of Statistical Mechanics: Theory and Experiment, 2014(10):P10036, 2014. Cited on p. 99.
• [13] K. Burnecki, A. Wylomanska, and A. Chechkin. Discriminating between light-and heavy-tailed distributions with limit theorem. PLoS One, 10(12):e0145604, 2015. Cited on p. 88.
• [14] S. Chan, J. Chu, S. Nadarajah, and J. Osterrieder. A statistical analysis of cryptocurrencies. Journal of Risk and Financial Management, 10(2), 2017. Cited on p. 111.
• [15] R. Chhikara. The Inverse Gaussian Distribution: Theory: Methodology, and Applications, volume 95. CRC Press, 1988. Cited on p. 91.
• [16] J. Davis. The crypto-currency. The New Yorker, 87, 2011. Cited on p. 85.
• [17] R. F. Engle. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society, pages 987-1007, 1982. Cited on p. 101.
• [18] C. L. Franzke. Persistent regimes and extreme events of the North Atlantic atmospheric circulation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1991):20110471, 2013. Cited on p. 112.
• [19] M. N. Giuricich and K. Burnecki. Modelling of left-truncated heavy-tailed data with application to catastrophe bond pricing. Physica A: Statistical Mechanics and Its Applications, 525:498-513, 2019. Cited on p. 88.
• [20] W. K. Härdle, C. R. Harvey, and R. C. Reule. Understanding cryptocurrencies. Journal of Financial Econometrics, 18(2):181-208, 2020. Cited on pp. 86 and 111.
• [21] A. J. Hou, W. Wang, C. Y.-H. Chen, and W. K. Härdle. Pricing cryptocurrency options: the case of Bitcoin and CRIX. Available at SSRN 3159130, 2019. Cited on p. 86.
• [22] S. Hurst. The Characteristic Function of the Student T Distribution. Financial mathematics research report. Centre for Mathematics and its Applications, School of Mathematical Sciences, ANU, 1995. Cited on p. 92.
• [23] J. P. Maheshchandra. Long memory property in return and volatility: Evidence from the Indian stock markets. Asian Journal of Finance & Accounting, 4(2):218-230, 2012. Cited on p. 86.
• [24] T. Kharrat, G. N. Boshnakov, and M. G. N. Boshnakov. Package ‘stableestim’. Package Version, 2, 2016. Cited on p. 88.
• [25] Y. Liu and A. Tsyvinski. Risks and returns of cryptocurrency. The Review of Financial Studies, 34(6):2689-2727, 2021. Cited on p. 86.
• [26] G. M. Ljung and G. E. Box. On a measure of lack of fit in time series models. Biometrika, 65(2):297-303, 1978. Cited on p. 94.
• [27] D. B. Madan, S. Reyners, and W. Schoutens. Advanced model calibration on bitcoin options. Digital Finance, 1:117-137, 2019. Cited on p. 86.
• [28] J. L. Matic, N. Packham, and W. K. Härdle. Hedging cryptocurrency options. Review of Derivatives Research, 26(1):91-133, 2023. Cited on p. 86.
• [29] R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1):1-77, 2000. Cited on p. 99.
• [30] S. Nadarajah. A generalized normal distribution. Journal of Applied Statistics, 32(7):685-694, 2005. Cited on p. 91.
• [31] J. P. Nolan. Univariate Stable Distributions. Models for Heavy Tailed Data. Springer, Cham, 2020. Cited on pp. 90 and 91.
• [32] M. Norton, V. Khokhlov, and S. Uryasev. Calculating cvar and bpoe for common probability distributions with application to portfolio optimization and density estimation. Annals of Operations Research, 299:1281-1315, 2021. Cited on p. 91.
• [33] W. Palma. Long-Memory Time Series: Theory and Methods. John Wiley & Sons, 2007. Cited on p. 98.
• [34] S. K. Panda, A. A. Elngar, V. E. Balas, and M. Kayed. Bitcoin and Blockchain: History and Current Applications. CRC Press, 2020. Cited on p. 85.
• [35] S. Rudkin, W. Rudkin, and P. Dłotko. On the topology of cryptocurrency markets. International Review of Financial Analysis, 89:102759, 2023. Cited on p. 86.
• [36] M. Swan. Blockchain: Blueprint for a New Economy. O’Reilly, 2015. Cited on p. 86.
• [37] P. P. Tan, D. U. Galagedera, and E. A. Maharaj. A wavelet based investigation of long memory in stock returns. Physica A: Statistical Mechanics and its Applications, 391:2330-2341, 2012. Cited on p. 86.
• [38] S. Taylor. Asset Price Dynamics, Volatility and Prediction. Princeton University Press, New Jersey, 2005. Cited on p. 104.
• [39] N. Tripathy. Long memory and volatility persistence across BRICS stock markets. Research in International Business and Finance, 63:101782, 2022. Cited on p. 86.
• [40] W. N. Venables and B. D. Ripley. Modern Applied Statistics with SPLUS. Springer, New York, 2013. Cited on p. 96.