Application of the Hurst coefficient, calculated with the use of the Siroky method on financial markets
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Published:
Dec 28, 2021
Keywords:
Hurst exponent, market efficiency, developed countries
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Abstract
The paper analyses the Hurst exponents calculated with the use of the Siroky method in two time intervals of 625 and 1250 sessions for the group of 570 financial instruments (Warsaw Stock Exchange equities – 320, equity indexes – 7, commodities – 41, and FX market – 135). The study also covers an analysis of the normality of the distribution of logarithmic rates of return, and the verification of statistical hypotheses with the use of the following statistical tests: Jarque-Bera (JB), Shapiro-Wilk (SW), and d’Agostino-Pearson (DA).
In the second part of the paper, the change of the Hurst coefficient over time was analysed, while in the third part two linear regressions of the form H(t) = a + m ∙ t were performed for each of the analysed assets, as well as the determination factor R2. This part of the study aims to answer the question whether the slope of the regression line has a positive or negative value and what the quality of such a fit is with the use of linear regression. Such an analysis enables to observe changes in the fractal dimension, and thus the risk in financial markets over a long period of time.
The main conclusion that was drawn from the research may be formulated as follows: the value of the H exponents decreased in the analysed time windows, which means an increase in the fractal dimension (d), and thus the investment risk in financial markets. The obtained results can be used in the process of constructing an investment portfolio in financial markets.
The research is part of the ongoing discussion on the effectiveness of financial markets.
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Borowski, K. ., & Matusewicz, M. (2021). Application of the Hurst coefficient, calculated with the use of the Siroky method on financial markets. Journal of Management and Financial Sciences, (42), 39–75. https://doi.org/10.33119/JMFS.2021.42.3
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References
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30. Kantelhardt, J., Zschiegner, S., Koscielny-Bunde, E., Budne, A., Havlin, S., Stanley, E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A., 1–4 (316).
31. Kale, M., Butar, F. (2011). Fractal analysis of time series and distribution properties of Hurst exponent. Journal of Mathematical Sciences & Mathematics Education, 5 (1), pp. 8–19.
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34. Kroha, P., Skoula, M. (2019). Hurst exponent and trading signals derived from market time series. Proceedings of the 20th International Conference on Enterprise Information Systems (ICEIS 2018). SCITEPRESS – Science and Technology Publications, pp. 371–378.
35. Kyaw, N., Los, C., Zong, S. (2004, 8 Nov). Persistence characteristics of Latin American financial markets. Kent State University Finance Working Paper. Retrieved from SSRN: https://ssrn.com/abstract=298745 or, pp. 1–34.
36. Lipka, J., Los, C. (2003). Long-term dependence characteristics of European stock indices. Economics Working Paper Archive, EconWPA, Finance Nº0409044. pp. 1–40.
37. Lo, A. (2004). The adaptive markets hypothesis: market efficiency from an evolutionary perspective. Journal of Portfolio Management, 30, pp. 15–29.
38. Mandelbrot, B., Wallis, J. (1968). Joah, Joseph and operational hydrology. Water-Resources Research, 4, pp. 909–918.
39. Mantenga, R., Stanley, H. (1995). Scaling behavior in the dynamics of an economic index. Nature, 376, pp. 46–49.
40. Marcinkiewicz, E. (2006). Badanie zależności pomiędzy wartością wykładnika Hursta a skutecznością strategii inwestycyjnych opartych na analizie technicznej. Zeszyty Naukowe SGGW, Ekonomika i Organizacja Gospodarki Żywnościowej, 60, pp. 231–239.
41. Mastalerz-Kodzis, A. (2003). Modelowanie procesów na rynku kapitałowym za pomocą multifraktali. Katowice: Wydawnictwo Akademii Ekonomicznej w Katowicach.
42. Muzy, J., Barcy, E., Arneodo, A. (1994). The Multifractal Formalism revisited with wavelets. National Journal of Bifurcation and Chaos, 2 (2), pp. 245–302.
43. Onali, E., Goddard, J. (2010). Are European equity markets efficient? New evidence from fractal analysis, pp. 59–67. Retrieved from SSRN: http://ssrn.com/abstract=1805044 [accessed: 12.03.2020].
44. Peng, C., Buldyrev, S., Havlin, S., Simons, M., Goldberger, A. (1994). Mosaic organization of DNA nucleotides. Physica Review E., 49 (2), pp. 1686–1689.
45. Peters, E. (1994). Fractal market analysis: applying chaos theory in investment and economics. New York: Wiley & Sons.
46. Peters, E. (1997). Teoria chaosu i rynki finansowe. Warszawa: WIG-PRESS.
47. Przekota, G. (2012). Szacowanie ryzyka zmian cen akcji metodą podziału pola. Problemy Zarządzania, 4 (10), pp. 186–187.
48. Raimundo, M., Okamoto, J. (2018). Application of Hurst exponent (H) and the R/S analysis in the classification of FOREX securities. International Journal of Modelling and Optimization, 8 (2), pp. 116–124.
49. Ryvkina, J. (2015). Fractional Brownian motion with variable Hurst parameter: definition and properties. Journal of Theretical Probability, 28, pp. 866–891.
50. Rzeszótko, Z. (2016). Analiza właściwości fraktalnych szeregów czasowych wybranych indeksów giełdowych. Metody Ilościowe w Badaniach Ekonomicznych, 3 (17), pp. 131–141.
51. Siroky, M. (2017). Estimating the fractal dimension on stock prices. Technical Analysis of Stock & Commodities, 12 (35), pp. 18–21 and 45.
52. Taqqu, M., Teverovsky, V. (1998). On estimating the intensity of long-range dependence in finite and infinite variance time series. Boston: Birkauser Boston Inc., pp. 177–217.
53. Veneziano, D. (1999). Basic properties and characterization of stochastically self-similar processes in R. Fractals, 7 (1), p. 60.
54. Voss R. (1991). Random fractal forgeries. R. Earnshaw (Ed.), Fundamental algorithms for computer graphics (pp. 816–817). Berlin: Springer.
55. Waściński, T., Przekota, G. (2012). Wybrane problemy oceny ryzyka zmian cen akcji za pomocą miar klasycznych i nieklasycznych. Zeszyty Naukowe Uniwersytetu Przyrodniczo-Humanistycznego w Siedlcach, 95, pp. 81–82.
56. Weron, A., Weron, R. (1998). Inżyniera finansowa. Warszawa Wydawnictwo Naukowo-Techniczne.
57. Wilder, W. (1978). New concept in technical trading systems. Greensboro: Trend Research.
58. Wołoszyn, J. (2001). Zjawisko błądzenia przypadkowego w ekonomicznych szeregach czasowych. Zeszyty naukowe Akademii Ekonomicznej w Krakowie, 569, pp. 5–23.
59. Zwolankowska, M. (2000a). Metoda segmentowo-wariacyjna. Nowa propozycja szacowania wymiaru fraktalnego. Przegląd Statystyczny, 1–2, pp. 209–224.
60. Zwolankowska, M. (2000b). Wymiar fraktalny jako miara zmienności stopy zwrotu. Zeszyty Naukowe Uniwersytetu Szczecińskiego, 306, p. 268.
61. Zwolankowska, M. (2001). Fraktalna geometria polskiego rynku akcji. Szczecin: Wydawnictwo Naukowe Uniwersytetu Szczecińskiego.
Internet websites:
1. http://ion.researchsystems.com/IONScript/wavelet/website [accessed: 12.11.2019].
2. Alessio, E., Carbone, A., Castelli, G., Frappietro, V. (2002). Second-order moving average and scaling of stochastic time series. European Physical Journal, 2 (27), pp. 197–200.
3. Alvarez-Ramirez, J., Echeverria, J., Rodriguez, E. (2008). Performance of a high-dimensional R/S method for Hurst exponent estimation. Physica A., 387, pp. 6452–6462.
4. Anis, A., Lloyd, E. (1976). The expected value of the adjusted Hurst range of independent normal summands. Biometrika, 63, pp. 111–116.
5. Barkoulas, J., Labys, W., Onochie, J. (1996). Fractional Dynamics In International Commodity Prices. Journal of Futures Markets, 17 (2), pp. 253–259.
6. Barunik, J., Kristoufek, L. (2010). On Hurst exponent estimation under heavy-tailed distributions. Physica A, 389, pp. 3844–3855.
7. Berg, L., Lyhagen, J. (1996). Short and long run dependence in Swedish stock returns, pp. 435–443. Retrieved from SSRN: http://ssrn.com/abstract=2270 [accessed: 12.03.2020].
8. Buła, R. (2015). Fluktuacje cen towarów rolnych w świetle analizy fraktalnej. In: B. Gołębiewska (Ed.), Wyzwania współczesnej gospodarki – aspekty teoretyczne i praktyczne (pp. 82–96). Warszawa: Wydawnictwo Szkoły Głównej Gospodarstwa Wiejskiego.
9. Buła, R. (2019). Implikacje teorii rynku fraktalnego dla oceny ryzyka inwestycji finansowych. Warszawa: Komisja Nadzoru Finansowego.
10. Bunde, A., Havlin, S. (1995). A brief introduction to fractal geometry. In: A. Bunde, S. Havlin (Eds.), Fractal and disordered systems (pp. 97–100). Berlin: Springer-Verlag.
11. Cajueiro, D., Tabak, B. (2004). Ranking efficiency for emerging markets. Chaos, Solitons and Fractals, 22, pp. 349–352.
12. Couillard, M., Davison, M. (2005). A comment on measuring the Hurst exponent on financial time series. Physica A, 348, pp. 404–418.
13. Corazza, M., Malliaris, A. (2008). Multifractality in foreign currency markets. Multinational Finance Journal, 6, pp. 387–401.
14. Czarnecka, A., Wilimowska, Z. (2018). Hurst exponent as a risk measurement on the capital market. In: J. Świątek, L. Borzemski, Z. Wilimowska Z. (Eds.), Information Systems Architecture and Technology. Proceedings of 38th International Conference on Information Systems Architecture and Technology, ISAT 2017. Warszawa: Springer International Publishing.
15. De La Fuente, I., Martinez, M., Aguirregabiria, J., Veguillas, J. (1998). R/S analysis in strange attractors. Fractals, 6 (2), pp. 95–100.
16. Di Matteo, T., Aste, T., Dacorogna, M. (2003). Scaling behaviors indifferently developed markets. Physica A., 324, pp. 183–188.
17. Dominique, C., Rivera, S. (2011). Mixed fractional Brownian motion, short and long-term dependence and economic conditions: the case of the S&P 500 index. International Business Management, 3, pp. 1–6.
18. Ehlers, J. (2005). Fractal Adaptive Moving Average. Technical Analysis of Stock & Commodities, 10 (23), pp. 81–82.
19. Ehlers, J., Way, R. (2010). Fractal dimension as a market mode sensor. Technical Analysis of Stock & Commodities, 6 (28), pp. 16–20.
20. Einstein, A. (1908). Elementare theorie der Brownschen bewungen, Zeitschrift für Elektrochemie und angewandte physikalische. Chemie, 14 (50), pp. 469–502.
21. Garcin, M. (2017). Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates. Physica A: Statistical Mechanics and its Applications, 483, pp. 462–479.
22. Glenn, L. (2007). On randomness and the NASDAQ Composite. Working Paper, pp. 1–19. Retrieved from SSRN: http://ssrn.com/abstract=1124991 [accessed: 12.03.2020].
23. Granger, C., Ding, Z. (1995). Some properties of absolute returns, an alternative measure of risk. Annales d’Economie et de Statistique, 40, pp. 67–91.
24. Granger, C., Hyungh, H. (2004). Occasional structural breaks and long memory with an application to S&P 500 absolute stock returns. Journal of Empirical Finance, 11, pp. 399–421.
25. Hannula, H. (1994). Polarized fractal efficiency. Technical Analysis of Stock & Commodities, 1 (12), pp. 38–41.
26. Henry, O. (2002). Long memory in stock returns, some international evidence. Applied Financial Economics, 12, pp. 725–729.
27. Huang, B., Yang, C. (1999). An examination of long-term memory using the intraday stock returns. Working Paper, Clarion University of Pennsylvania, pp. 1–19.
28. Hurst, H. (1951). Long term storage capacity of reservoirs. Transactions of American Society of Civil Engineers, 116, pp. 770–799.
29. Jacobsen, B. (1995). Are stock returns long term dependent? Some empirical evidence. Journal of International Financial Markets, Institutions and Money, 5 (2/3), pp. 37–52.
30. Kantelhardt, J., Zschiegner, S., Koscielny-Bunde, E., Budne, A., Havlin, S., Stanley, E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A., 1–4 (316).
31. Kale, M., Butar, F. (2011). Fractal analysis of time series and distribution properties of Hurst exponent. Journal of Mathematical Sciences & Mathematics Education, 5 (1), pp. 8–19.
32. Kapecka, A. (2015). Analiza porównawcza wybranych indeksów giełdowych rynków dojrzałych i wschodzących z wykorzystaniem wykładnika Hursta. Acta Universitatis Nicolai Copernici, Ekonomia, 1 (46), pp. 59–75.
33. Kowagier, H. (2009). Kilka uwag o wymiarze fraktalnym Minkowskiego oraz wykładniku Hursta na Giełdzie Papierów Wartościowych. Studia i Prace Wydziału Nauk Ekonomicznych i Zarządzania, 15, pp. 157–167.
34. Kroha, P., Skoula, M. (2019). Hurst exponent and trading signals derived from market time series. Proceedings of the 20th International Conference on Enterprise Information Systems (ICEIS 2018). SCITEPRESS – Science and Technology Publications, pp. 371–378.
35. Kyaw, N., Los, C., Zong, S. (2004, 8 Nov). Persistence characteristics of Latin American financial markets. Kent State University Finance Working Paper. Retrieved from SSRN: https://ssrn.com/abstract=298745 or, pp. 1–34.
36. Lipka, J., Los, C. (2003). Long-term dependence characteristics of European stock indices. Economics Working Paper Archive, EconWPA, Finance Nº0409044. pp. 1–40.
37. Lo, A. (2004). The adaptive markets hypothesis: market efficiency from an evolutionary perspective. Journal of Portfolio Management, 30, pp. 15–29.
38. Mandelbrot, B., Wallis, J. (1968). Joah, Joseph and operational hydrology. Water-Resources Research, 4, pp. 909–918.
39. Mantenga, R., Stanley, H. (1995). Scaling behavior in the dynamics of an economic index. Nature, 376, pp. 46–49.
40. Marcinkiewicz, E. (2006). Badanie zależności pomiędzy wartością wykładnika Hursta a skutecznością strategii inwestycyjnych opartych na analizie technicznej. Zeszyty Naukowe SGGW, Ekonomika i Organizacja Gospodarki Żywnościowej, 60, pp. 231–239.
41. Mastalerz-Kodzis, A. (2003). Modelowanie procesów na rynku kapitałowym za pomocą multifraktali. Katowice: Wydawnictwo Akademii Ekonomicznej w Katowicach.
42. Muzy, J., Barcy, E., Arneodo, A. (1994). The Multifractal Formalism revisited with wavelets. National Journal of Bifurcation and Chaos, 2 (2), pp. 245–302.
43. Onali, E., Goddard, J. (2010). Are European equity markets efficient? New evidence from fractal analysis, pp. 59–67. Retrieved from SSRN: http://ssrn.com/abstract=1805044 [accessed: 12.03.2020].
44. Peng, C., Buldyrev, S., Havlin, S., Simons, M., Goldberger, A. (1994). Mosaic organization of DNA nucleotides. Physica Review E., 49 (2), pp. 1686–1689.
45. Peters, E. (1994). Fractal market analysis: applying chaos theory in investment and economics. New York: Wiley & Sons.
46. Peters, E. (1997). Teoria chaosu i rynki finansowe. Warszawa: WIG-PRESS.
47. Przekota, G. (2012). Szacowanie ryzyka zmian cen akcji metodą podziału pola. Problemy Zarządzania, 4 (10), pp. 186–187.
48. Raimundo, M., Okamoto, J. (2018). Application of Hurst exponent (H) and the R/S analysis in the classification of FOREX securities. International Journal of Modelling and Optimization, 8 (2), pp. 116–124.
49. Ryvkina, J. (2015). Fractional Brownian motion with variable Hurst parameter: definition and properties. Journal of Theretical Probability, 28, pp. 866–891.
50. Rzeszótko, Z. (2016). Analiza właściwości fraktalnych szeregów czasowych wybranych indeksów giełdowych. Metody Ilościowe w Badaniach Ekonomicznych, 3 (17), pp. 131–141.
51. Siroky, M. (2017). Estimating the fractal dimension on stock prices. Technical Analysis of Stock & Commodities, 12 (35), pp. 18–21 and 45.
52. Taqqu, M., Teverovsky, V. (1998). On estimating the intensity of long-range dependence in finite and infinite variance time series. Boston: Birkauser Boston Inc., pp. 177–217.
53. Veneziano, D. (1999). Basic properties and characterization of stochastically self-similar processes in R. Fractals, 7 (1), p. 60.
54. Voss R. (1991). Random fractal forgeries. R. Earnshaw (Ed.), Fundamental algorithms for computer graphics (pp. 816–817). Berlin: Springer.
55. Waściński, T., Przekota, G. (2012). Wybrane problemy oceny ryzyka zmian cen akcji za pomocą miar klasycznych i nieklasycznych. Zeszyty Naukowe Uniwersytetu Przyrodniczo-Humanistycznego w Siedlcach, 95, pp. 81–82.
56. Weron, A., Weron, R. (1998). Inżyniera finansowa. Warszawa Wydawnictwo Naukowo-Techniczne.
57. Wilder, W. (1978). New concept in technical trading systems. Greensboro: Trend Research.
58. Wołoszyn, J. (2001). Zjawisko błądzenia przypadkowego w ekonomicznych szeregach czasowych. Zeszyty naukowe Akademii Ekonomicznej w Krakowie, 569, pp. 5–23.
59. Zwolankowska, M. (2000a). Metoda segmentowo-wariacyjna. Nowa propozycja szacowania wymiaru fraktalnego. Przegląd Statystyczny, 1–2, pp. 209–224.
60. Zwolankowska, M. (2000b). Wymiar fraktalny jako miara zmienności stopy zwrotu. Zeszyty Naukowe Uniwersytetu Szczecińskiego, 306, p. 268.
61. Zwolankowska, M. (2001). Fraktalna geometria polskiego rynku akcji. Szczecin: Wydawnictwo Naukowe Uniwersytetu Szczecińskiego.
Internet websites:
1. http://ion.researchsystems.com/IONScript/wavelet/website [accessed: 12.11.2019].