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Nonlinear stochastic exclusion financial dynamics modeling and complexity behaviors

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Abstract

In attempt to reproduce and investigate nonlinear dynamics of financial markets, a new random agent-based financial price dynamics is developed and investigated by stochastic exclusion process. The exclusion process, one of Markov interacting processes, is firstly introduced to imitate the trading interactions among the investing agents in this work and to explain various statistical facts found in financial data. To better understand the fluctuation complexity properties of the proposed model, the complex analyses of random logarithmic price return series are preformed, including power-law distribution, Lempel–Ziv complexity, correlation dimension analysis, maximum Lyapunov exponent, mean Lyapunov exponents and Kolmogorov–Sinai entropy density. In order to verify the rationality of the model, the corresponding analyses of real return series are also studied for comparison. The empirical research reveals that this financial model can reproduce similar statistical behaviors, power-law distribution of returns, complexity and chaotic features of returns for real stock markets.

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References

  1. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Duarte, F.B., Machado, J.A.T., Duarte, G.M.: Dynamics of the Dow Jones and the NASDAQ stock indexes. Nonlinear Dyn. 61, 691–705 (2010)

    Article  MATH  Google Scholar 

  4. Elliott, R.J., Siu, T.K., Fung, E.S.: Filtering a nonlinear stochastic volatility model. Nonlinear Dyn. 67, 1295–1313 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lux, T.: Financial Power Laws: Empirical Evidence, Models and Mechanisms. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  6. Machado, J.A.T.: Complex dynamics of financial indices. Nonlinear Dyn. 74, 287–296 (2013)

    Article  Google Scholar 

  7. Machado, J.A.T., Duarte, F.B., Duarte, G.M.: Analysis of financial data series using fractional Fourier transform and multidimensional scaling. Nonlinear Dyn. 65, 235–245 (2011)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Gaylord, R., Wellin, P.: Computer Simulations with Mathematica: Explorations in the Physical, Biological and Social Science. Springer, New York (1995)

    MATH  Google Scholar 

  10. Ilinski, K.: Physics of Finance, Gauge Modeling in Nonequilibrium Pricing. Wiley, New York (2001)

    Google Scholar 

  11. Liu, F.J., Wang, J.: Fluctuation prediction of stock market index by Legendre neural network with random time strength function. Neurocomputing 83, 12–21 (2012)

    Article  Google Scholar 

  12. Mills, T.C.: The Econometric Modeling of Financial Time Series, 2nd edn. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Tavares, A.B., Curto, J.D., Tavares, G.N.: Modelling heavy tails and asymmetry using ARCH-type models with stable Paretian distributions. Nonlinear Dyn. 51, 231–243 (2008)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  16. Yu, Y., Wang, J.: Lattice-oriented percolation system applied to volatility behavior of stock market. J. Appl. Stat. 39, 785–797 (2012)

    Article  MathSciNet  Google Scholar 

  17. Yang, G., Wang, J., Fang, W.: Numerical analysis for finite-range multitype stochastic contact financial market dynamic systems. Chaos 25, 043111 (2015)

    Article  MathSciNet  Google Scholar 

  18. Fang, W., Wang, J.: Fluctuation behaviors of financial time series by a stochastic Ising system on a Sierpinski carpet lattice. Phys. A 392, 4055–4063 (2013)

    Article  Google Scholar 

  19. Niu, H.L., Wang, J.: Complex dynamic behaviors of oriented percolation-based financial time series and Hang Seng index. Chaos Solitons Fractals 52, 36–44 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  21. Yang, G., Chen, Y., Huang, J.P.: The highly intelligent virtual agents for modeling financial markets. Phys. A 443, 98–108 (2016)

    Article  Google Scholar 

  22. Shang, Y.: An agent based model for opinion dynamics with random confidence threshold. Commun. Nonlinear Sci. 19, 3766–3777 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Vladimirov, S.N.: Probabilistic method of calculating the Maximum Lyapunov Exponent for motion of chaotic systems. Russ. Phys. J. 48, 660–663 (2005)

    Article  Google Scholar 

  25. Awrejcewicz, J., Krysko, V.A., Kutepov, I.E., Vygodchikova, I.Y., Krysko, A.V.: Quantifying chaos of curvilinear beams via exponents. Commun. Nonlinear Sci. 27, 81–92 (2015)

    Article  MathSciNet  Google Scholar 

  26. Piqueira, J.R.C., Mortoza, L.P.D.: Brazilian exchange rate complexity: financial crisis effects. Commun. Nonlinear Sci. 17, 1690–1695 (2012)

    Article  Google Scholar 

  27. Zhang, Y.Q., Wang, X.Y.: A new image encryption algorithm based on non-adjacent coupled map lattices. Appl. Soft Comput. 26, 10–20 (2015)

    Article  Google Scholar 

  28. Zhang, Y.Q., Wang, X.Y.: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Gerodimos, N.A., Daltzis, P.A., Hanias, M.P., Nistazakis, H.E., Tombras, G.S.: Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Lorenz Maps. Springer, Berlin (2013)

    Book  Google Scholar 

  31. Niu, H.L., Wang, J.: Quantifying complexity of financial short-term time series by composite multiscale entropy measure. Commun. Nonlinear Sci. 22, 375–382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985)

    Book  MATH  Google Scholar 

  34. Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York (1999)

    Book  MATH  Google Scholar 

  35. Sopasakis, A., Katsoulakis, M.A.: Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with arrhenius look-ahead dynamics. SIAM J. Appl. Math. 66, 921–944 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gray, L., Griffeath, D.: The ergodic theory of traffic jams. J. Stat. Phys. 105, 413–452 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Fernández, A., López-Ibor, M., Turrero, A., Santos, J., Morón, M., Hornero, R., Gómez, C., Méndez, M.A., Ortiz, T., López-lbor, J.J.: Lempel–Ziv complexity in shizophrenia: a MEG study. Clin. Neurophysiol. 122, 2227–2235 (2011)

    Article  Google Scholar 

  39. Abásolo, D., Hornero, R., Gómez, C., García, M., López, M.: Analysis of EEG background activity in Alzheimer’s disease patients with Lempel–Ziv complexity and central tendency measure. Med. Eng. Phys. 28, 315–322 (2006)

    Article  Google Scholar 

  40. 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)

    Article  Google Scholar 

  41. Tang, J., Wang, Y., Wang, H., Zhang, S., Liu, F.: Dynamic analysis of traffic time series at different temporal scales: a complex networks approach. Phys. A 405, 303–315 (2014)

    Article  Google Scholar 

  42. Zhang, X.S., Roy, R.J., Jensen, E.W.: EEG complexity as a measure of depth of anesthesia for patients. IEEE Trans. Biomed. Eng. 48, 1424–1433 (2001)

    Article  Google Scholar 

  43. Yan, R., Gao, R.X.: Complexity as a measure for machine health evaluation. IEEE Trans. Instrum. Measu. 53, 1327–1334 (2004)

    Article  Google Scholar 

  44. Wu, X., Xu, J.: Complexity and brain function. Acta Biophys. Sin. 7, 103–106 (1991)

    Google Scholar 

  45. Yilmaz, D., Gler, N.F.: Analysis of the Doppler signals using largest Lyapunov exponent and correlation dimension in healthy and stenosed internal carotid artery patients. Digit. Signal Process. 20, 401–409 (2010)

    Article  Google Scholar 

  46. Takens, F.: Detecting Strange Attractors in Turbulence. Springer, New York (1981)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Grassberger, P., Proccacia, I.: Measuring the strangeness of strange attractors. Phys. D 9, 189–208 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  49. Rosenstein, M.T., Collins, J.J., DeLuca, C.J.: Apractical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)

  50. Brigham, E.O.: The Fast Fourier Transform. Prentice-hall Inc, Englewood (1974)

    MATH  Google Scholar 

  51. Shibata, H.: KS entropy and mean Lyapunov exponent for coupled map lattices. Phys. A 292, 182–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors were supported in part by National Natural Science Foundation of China Grant No. 71271026.

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  1. School of Science, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Wei Zhang & Jun Wang

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  1. Wei Zhang
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  2. Jun Wang
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Correspondence to Wei Zhang.

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Zhang, W., Wang, J. Nonlinear stochastic exclusion financial dynamics modeling and complexity behaviors. Nonlinear Dyn 88, 921–935 (2017). https://doi.org/10.1007/s11071-016-3285-0

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