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Complexity behaviors of volatility dynamics for stochastic Potts financial model

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Abstract

To investigate the price fluctuation mechanism of stock markets, this research aims to develop a novel stochastic financial model based on Potts dynamics and compound Poisson process. The new model considers two aspects: information interaction among traders and the uncertain events outside the system. Then, three different volatility statistics (return series \(r_t\), absolute return series \(|r_t|\) and volatility duration average intensity \(V_t\)) are introduced to explore the volatility and complexity properties of the proposed model. The descriptive statistical methods, such as basic statistical properties and distribution analysis, are studied to validate the practicable of the proposed stochastic financial model. The permutation Lempel-Ziv complexity of moving average series is referred to different volatility sequences to evaluate the complexity of the simulative data from proposed model and the real data from stock market. Moreover, the complexity analysis of fractional sample entropy and multiscale fractional sample entropy is improved to illustrate the complexity of volatility behaviors in different scales. Compared with the real stock data, the empirical results demonstrate that the new model could reproduce the fluctuation and volatility behaviors of real stock markets to some extent.

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Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Farmer, J.D., Joshi, S.: The price dynamics of common trading strategies. J. Econ. Behav. Organ. 49, 149–171 (2002)

    Google Scholar 

  2. Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multiagent model of a financial market. Nature. 397, 498–500 (1999)

    Google Scholar 

  3. Bouchaud, J.P., Potters, M.: Theory of Financial risk and derivative pricing: from statistical physics to risk management. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  4. Stanley, H.E., Gabaix, X., Gopikrishnan, P., Plerou, V.: Economic fluctuations and statistical physics: the puzzle of large fluctuations. Nonlinear Dynam. 44, 329–340 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Gontis, V., Havlina, S., Kononovicius, A., et al.: Stochastic model of financial markets reproducing scaling and memory in volatility return intervals. Phyisca A. 462, 1091–1102 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Gabaix, X., Gopikrishanan, P., Plerou, V., Stanley, H.E.: A theory of power-law distributions in financial market fluctuations. Nature. 423, 267–270 (2003)

    Google Scholar 

  7. Calvet, L., Fisher, A.: Multifractal Volatility: Theory, Forecasting, and Pricing. Academic Press, New York, USA (2008)

    MATH  Google Scholar 

  8. Grech, D., Pamula, G.: Multifractality of nonlinear transformations with application in finances. Acta Phys. Pol. A. 123, 529–537 (2013)

    Google Scholar 

  9. Amaral, L.A.N., Buldyrev, S.V., Havlin, S., et al.: Power law scaling for a system of interacting units with complex internal structure. Phys. Rev. Lett. 80, 1385–1388 (1998)

    Google Scholar 

  10. Wang, J., Wang, J.: Cross-correlation complexity and synchronization of the financial time series on Potts dynamics. Physica A. (2020). https://doi.org/10.1016/j.physa.2019.123286

    Article  Google Scholar 

  11. Li, J., Shang, P., Zhang, X.: Financial time series analysis based on fractional and multiscale permutation entropy. Communications in Nonlinear Science and Numerical Simulation. 78, 104880 (2019)

  12. Lux, T.: Estimation of agent-based models using sequential Monte Carlo methods. J. Econ. Dyn. Control. 91, 391–408 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Boswijk, H.P., Hommes, C.H., Manzan, S.: Behavioral heterogeneity in stock prices. J. Econ. Dyn. Control. 31(6), 1938–1970 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Durrett, R.: Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks, Pacific Grove-California (1998)

    MATH  Google Scholar 

  15. Zivot, E., Wang, J.H.: Modeling Financial Time Series with SPLUS. Springer, New York (2006)

    MATH  Google Scholar 

  16. Mike, S., Farmer, J.D.: An empirical behavioral model of liquidity and volatility. J. Econ. Dyn. Control. 32, 200–234 (2008)

    Google Scholar 

  17. Stauffer, D., Penna, T.J.P.: Crossover in the Cont-Bouchaud percolation model for market fluctuations. Physica A. 256, 284–290 (1998)

    Google Scholar 

  18. Wang, J., Wang, J., Stanley, H.E.: Multiscale multifractal DCCA and complexity behaviors of return intervals for Potts price model. Physica A. 492, 889–902 (2018)

    MathSciNet  Google Scholar 

  19. Wang, Y., Zheng, S., Zhang, W., et al.: Modeling and complexity of stochastic interacting Lévy type financial price dynamics. Physica A. 499, 498–511 (2018)

    MathSciNet  Google Scholar 

  20. Zhang, B., Wang, G., Wang, Y., et al.: Multiscale statistical behaviors for Ising financial dynamics with continuum percolation jump. Physica A. 525, 1012–1025 (2019)

    Google Scholar 

  21. Wang, J., Wang, J.: Measuring the correlation complexity between return series by multiscale complex analysis on Potts dynamics. Nonlinear Dyn. 89, 2703–2721 (2017)

    MathSciNet  Google Scholar 

  22. Bornholdt, S., Wagner, F.: Stability of money: phase transitions in an Ising economy. Physica A. 316, 453–468 (2002)

    MATH  Google Scholar 

  23. Wang, F., Yamasaki, K., Havlin, S., Stanley, H.E.: Scaling and memory of intraday volatility return intervals in stock markets. Phys. Rev. E. (2006). https://doi.org/10.1103/PhysRevE.73.026117

    Article  Google Scholar 

  24. Yang, G., Wang, J., Deng, W.: Nonlinear analysis of volatility duration financial series model by stochastic interacting dynamic system. Nonlinear Dyn. 80(1), 701–713 (2015)

    MathSciNet  Google Scholar 

  25. Wang, H., Wang, J., Wang, G.: Nonlinear continuous fluctuation intensity financial dynamics and complexity behavior. Chaos: An Interdisciplinary Journal of Nonlinear Science. 28(8), 083122 (2018)

  26. Li, R., Wang, J.: Symbolic complexity of volatility duration and volatility difference component on voter financial dynamics. Digital Signal Process. 63, 56–71 (2017)

    Google Scholar 

  27. Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E. (2005). https://doi.org/10.1103/PhysRevE.71.021906

    Article  MathSciNet  Google Scholar 

  28. Wu, Y., Shang, P., Li, Y.: Multiscale sample entropy and crosssample entropy based on symbolic representation and similarity of stock markets. Commun. Nonlinear Sci. Num. Simul. 56, 49–61 (2018)

    MATH  Google Scholar 

  29. Hong, H., Liang, M.: Fault severity assessment for rolling element bearings using the Lempel-Ziv complexity and continuous wavelet transform. J. Sound Vib. 320, 452–468 (2009)

    Google Scholar 

  30. Fernández, A., LópezIbor, M.I., Turrero, A., et al.: Lempel-Ziv complexity in schizophrenia: a MEG study. Clin. Neurophys. 122, 2227–2235 (2011)

    Google Scholar 

  31. Bai, Y., Liang, Z., Li, X.: A permutation Lempel-Ziv complexity measure for EEG analysis. Biomed. Signal Process. Control. 19, 102–114 (2015)

    Google Scholar 

  32. Xu, K., Wang, J.: Weighted fractional permutation entropy and fractional sample entropy for nonlinear Potts financial dynamics. Phys. Lett. A 381(8), 767–779 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Lempel, A., Ziv, J.: On the complexity of finite sequences. Inform. Theory IEEE Trans. 22, 75–81 (1976)

    MathSciNet  MATH  Google Scholar 

  34. Potts, R.: Some generalized order-disorder transformations. Math. Proc. Cambr. Philos. Soc. 48(1), 106–109 (1952)

    MathSciNet  MATH  Google Scholar 

  35. Wu, F.Y.: The potts model. Rev. Modern Phys. 54(1), 235 (1982)

    MathSciNet  Google Scholar 

  36. Gliozzi, F.: Simulation of potts models with real \(q\) and no critical slowing down. Phys. Rev. E. (2002). https://doi.org/10.1103/PhysRevE.66.016115

    Article  Google Scholar 

  37. Deng, Y., Blóte, H.W.J., Nienhuis, B.: Backbone exponents of the two-dimensional \(q\)-state Potts model: a Monte Carlo investigation. Phys. Rev. E. (2004). https://doi.org/10.1103/PhysRevE.69.026114

    Article  Google Scholar 

  38. Cont, R., Tankov, P.: Financial modeling with jump processes. Chapman & HallI CRC, USA (2004)

    MATH  Google Scholar 

  39. Tosun, P.D., Abásolo, D., Stenson, G., et al.: Characterisation of the Effects of Sleep Deprivation on the Electroencephalogram Using Permutation Lempel-Ziv Complexity, a Non-Linear Analysis Tool. Entropy 19(12), 673 (2017)

    Google Scholar 

  40. Baxter, R.J.: Exactly solved models in statistical mechanics. Elsevier, Amsterdam (2016)

    MATH  Google Scholar 

  41. Tsay, R.S.: Analysis of Financial Time Series. Wiley, Hoboken, USA (2005)

    MATH  Google Scholar 

  42. Ross, S.M.: An Introduction to Mathematical Finance. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  43. Mantegna, R.N., Stanley, H.E.: An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  44. Tangmongkollert, K., Suwanna, S.: Modeling of price and profit in coupled-ring networks. Eur. Phys. J. B. 89, 146 (2016)

    MATH  Google Scholar 

  45. Podobnik, B., Fu, D.F., Stanley, H.E., Ivanov, P.C.: Power-law auto-correlated stochastic processes with long-range crosscorrelations. Eur. Phys. J. B. 56, 47–52 (2007)

    Google Scholar 

  46. Weber, M.D., Leemis, L.M., Kincaid, R.K.: Minimum Kolmogorov- Smirnov test statistic parameter estimates. J. Statist. Comput. Simul. 76(3), 195–206 (2006)

    MathSciNet  MATH  Google Scholar 

  47. Gao, Z., Jin, N.: Complex network from time series based on phase space reconstruction. Chaos: An Interdisciplinary Journal of Nonlinear Science 19(3), 033137 (2009)

  48. Han, L., Romero, C.E., Yao, Z.: Wind power forecasting based on principle component phase space reconstruction. Renew. Energy 81, 737–744 (2015)

    Google Scholar 

  49. Takens, F.: Detecting strange attractors inturbulence. Springer, Berlin Heidelberg (1981)

    MATH  Google Scholar 

  50. Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45, 3403 (1992)

    Google Scholar 

  51. Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140 (1986)

    MathSciNet  MATH  Google Scholar 

  52. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D Nonlinear Phenom. 65(1–2), 117–134 (1993)

    MathSciNet  MATH  Google Scholar 

  53. Wolf, A., Swift, J.B., Swinney, H.L., et al.: Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 16(3), 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  54. Shibata, H.: KS entropy and mean Lyapunov exponent for coupled map lattices. Phys. A Statist. Mech. Appl. 292(1–4), 182–192 (2001)

    MathSciNet  MATH  Google Scholar 

  55. Lim, J.H., Khang, E.J., Lee, T.H., et al.: Detrended fluctuation analysis and Kolmogorov-Sinai entropy of electroencephalogram signals. Phys. Lett. A 377(38), 2542–2545 (2013)

    Google Scholar 

  56. Zhang, Y.Q., Wang, X.Y.: Spatiotemporal chaos in mixed linear-nonlinear coupled logistic map lattice. Phys. A 402, 104–118 (2014)

    MathSciNet  MATH  Google Scholar 

  57. Li, R., Wang, J.: Interacting price model and fluctuation behavior analysis from Lempel-Ziv complexity and multi-scale weightedpermutation entropy. Phys. Lett. A. 380, 117–129 (2016)

    MathSciNet  Google Scholar 

  58. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. (2002). https://doi.org/10.1103/PhysRevLett.88.174102

    Article  Google Scholar 

  59. Machado, J.A.T.: Fractional order generalized information. Entropy 16(4), 2350–2361 (2014)

    Google Scholar 

  60. Lopes, A.M., Machado, J.A.T.: Multidimensional scaling analysis of generalized mean discrete-time fractional order controllers. Commun. Nonlinear Sci. Num. Simul. (2021). https://doi.org/10.1016/j.cnsns.2020.105657

    Article  MathSciNet  MATH  Google Scholar 

  61. Richman, J.S., Lake, D.E., Moorman, J.R.: Sample entropy. Methods Enzymol. 384, 172–184 (2004)

    Google Scholar 

  62. Baleanu, D., Diethelm, K., Scalas, E., et al.: Fractional calculus: models and numerical methods. World Scientific, Singapore (2012)

    MATH  Google Scholar 

  63. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles. Fields and Media. Springer, Dordrecht (2010)

    MATH  Google Scholar 

  64. Machado, J.A.T., Silva, M.F., Barbosa, R.S., et al.: Some applications of fractional calculus in engineering. Math. Prob Eng. 2010,(2010)

  65. Momani, S., Abu Arqub, O., Maayah, B.: Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu model of the Lienard’s equation. Fractals 28(8), 2040007 (2020)

    Google Scholar 

  66. Momani, S., Arqub, O.A., Maayah, B.: The reproducing kernel algorithm for numerical solution of Van der Pol damping model in view of the Atangana-Baleanu fractional approach. Fractals 28(8), 2040010 (2020)

    Google Scholar 

  67. Arqub, O.A.: Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates. J. Appl. Math. Comput. 59(1), 227–243 (2019)

    MathSciNet  MATH  Google Scholar 

  68. Arqub, O.A., Rashaideh, H.: The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs. Neural Comput. Appl. 30(8), 2595–2606 (2018)

    Google Scholar 

  69. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, Hoboken (1993)

  70. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  71. Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373(30), 2516–2519 (2009)

    MathSciNet  MATH  Google Scholar 

  72. Zunino, L., Pérez, D.G., Martín, M.T., Garavaglia, M., Plastino, A., Rosso, O.A.: Permutation entropy of fractional Brownian motion and fractional Gaussian noise. Phys. Lett. A. 372, 4768–4774 (2008)

    MATH  Google Scholar 

  73. Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of complex physiological time series. Phys. Rev. Lett. (2002). https://doi.org/10.1103/PhysRevLett.89.068102

    Article  Google Scholar 

  74. Thuraisingham, R.A., Gottwald, G.A.: On multiscale entropy analysis for physiological data. Phys. A. 366, 323–332 (2006)

    Google Scholar 

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Acknowledgements

This research was supported by North China University of Technology (Grant No.110051360002).

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  1. Department of Statistics, College of Science, North China University of Technology, Beijing, 100144, People’s Republic of China

    Jie Wang

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  1. Jie Wang
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Wang, J. Complexity behaviors of volatility dynamics for stochastic Potts financial model. Nonlinear Dyn 105, 1097–1119 (2021). https://doi.org/10.1007/s11071-021-06593-y

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