Skip to main content
SpringerLink
Log in

The Inherent Law of the Unpredictability of Financial Asset Price Fluctuations: Multistability and Chaos

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper aims at understanding the price dynamics generated by the interaction of traders relying on heterogeneous expectations in an asset pricing model. In the present work the authors analyze a financial market populated by five types of boundedly rational speculators-two types of fundamentalists, two types of chartists and trend followers which submit buying/selling orders according to different trading rules. The authors formulate a stock market model represented as a 2 dimensional piecewise linear discontinuous map. The proposed contribution to the existing financial literature is two aspects. First, the authors perform study of the model involving a 2 dimensional piecewise linear discontinuous map through a combination of qualitative and quantitative methods. The authors focus on the existence conditions of chaos and the multi-stability regions in parameter plane. Related border collision bifurcation curves and basins of multi-attractors are also given. The authors find that chaos or quasi-period exists only in the case of fixed point being a saddle (regular or flip) and that the coexistence of multiple attractors may exist when the fixed point is an attractor, but it is common for spiral and flip fixed points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hommes A C, Evolutionary selection of individual expectations and aggregate outcomes in asset pricing experiments, American Econ. J.: Microeconomics, 2012, 4(4): 35–64.

    MathSciNet  Google Scholar 

  2. Brock W and Hommes C, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, J. of Econ. Dyn. and Control, 1998, 22: 1235–1274.

    Article  MathSciNet  Google Scholar 

  3. Chiarella C, Dieci R, and He X Z, Heterogeneity, market mechanisms, and asset price dynamics, Eds. by Hens T and Schenk-Hopp K R, Handbook of Financial Markets: Dynamics and Evolution, North-Holland, Amsterdam, 2009.

    Google Scholar 

  4. Hommes C, Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge University Press, Cambridge, U.K., 2013.

    Book  Google Scholar 

  5. Tramontana F, Gardini L, and Puu T, Duopoly games with alternative technologies, J. Econ. Dyn. Control, 2009, 33: 250–265.

    Article  Google Scholar 

  6. Westerhoff F, Exchange rate dynamics: A nonlinear survey, Ed. by Rosser J B, Handbook of Research on Complexity, Cheltenham, Edward Elgar, 2009.

    Google Scholar 

  7. Brock W A, Hommes C H, and Wagener F, More hedging instruments may destabilize markets, J. of Econ. Dyn. Control, 2009, 33(11): 1912–1928.

    Article  MathSciNet  Google Scholar 

  8. Lux T, Stochastic Behavioural Asset-Pricing Models And The Stylized Facts, Eds. by Hens T, Schenk-Hopp K R, Handbook of Financial Markets: Dynamics and Evolution, North Holland, Amsterdam, 2009, 161–216.

    Chapter  Google Scholar 

  9. LeBaron B, Agent-based computational finance, Eds. by Tesfatsion L and Judd K, Handbook of Computational Economics, Agent-Based Computational Economics, North Holland, 2006, 2: 1187–1233.

  10. Naimzada A K and Ricchiuti G, Heterogeneous fundamentalists and imitative processes, Appl. Math Comput., 2008, 199(1): 171–80.

    MathSciNet  Google Scholar 

  11. Naimzada A K and Ricchiuti G, Dynamic effects of increasing heterogeneity in financial markets, Chaos Solitons & Fractal, 2009, 41(4): 1764–72.

    Article  Google Scholar 

  12. Westerhoff F, Multiasset market dynamics, Macroecon. Dyn., 2004, 8(5): 96–616.

    Article  Google Scholar 

  13. Agliari A, Naimzada A, and Pecora N, Boom-bust dynamics in a stock market participation model with heterogeneous traders, J Econ. Dyn. Control, 2018, 91: 458–68.

    Article  MathSciNet  Google Scholar 

  14. Hommes C and LeBaron B, Computational Economics: Heterogeneous Agent Modeling, Elsevier, 2018.

  15. Polach J and Kukacka J, Prospect theory in the heterogeneous agent model, J. Econ. Interact. Coordin., 2019, 14(1): 147–174.

    Article  Google Scholar 

  16. Ellen S T and Verschoor W, Heterogeneous beliefs and asset price dynamics: A survey of recent evidence, Uncertainty, Expectations and Asset Price Dynamics, Springer, 2018, 53–79.

  17. Jahanshahi H, Sajjadi S S, Bekiros S, et al., On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller, Chaos Solitons & Fractals, 2021, 144: 110698.

    Article  MathSciNet  Google Scholar 

  18. Jahanshahi H, Yousefpour A, Wei Z, et al., A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization, Chaos, Solitons & Fractals, 2019, 126: 66–77.

    Article  MathSciNet  Google Scholar 

  19. Ay A, Hj B, Mp C, et al., A fractional-order hyper-chaotic economic system with transient chaos, Chaos, Solitons & Fractals, 2020, 130(C): 109400–109400.

    MathSciNet  Google Scholar 

  20. Day R and Huang W, Bulls, bears and market sheep, J. Econ. Behav. Org., 1990, 14: 299–329.

    Article  Google Scholar 

  21. Day R, Complex dynamics, market mediation and stock price behavior, North American Actuarial J., 1997, 1: 6–21.

    Article  MathSciNet  Google Scholar 

  22. DeGrauwe P, Dewachter H, and Embrechts M, Exchange Rate Theories, Chaotic Models of the Foreign Exchange Market, Oxford, Blackwell, 1993.

    Google Scholar 

  23. Farmer J D and Joshi S, The price dynamics of common trading strategies, J. of Econ. Behavior and Org., 2002, 49: 149–171.

    Article  Google Scholar 

  24. Tramontana F, Westerhoff F, and Gardini L, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders, J. of Econ. Beha. and Org., 2010, 74: 187–205.

    Article  Google Scholar 

  25. Tramontana F, Westerhoff F, and Gardini L, The bull and bear market model of Huang and Day: Some extensions and new results, J. of Econ. Dyn. and Control, 2013, 37: 2351–2370.

    Article  MathSciNet  Google Scholar 

  26. Tramontana F, Westerhoff F, and Gardini L, One-dimensional maps with two discontinuity points and three linear branches: Mathematical lessons for understanding the dynamics of financial markets, Decisions in Econ. and Finance, 2014, 37: 27–51.

    Article  MathSciNet  Google Scholar 

  27. Tramontana F, Westerhoff F, and Gardini L, A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities, Math. and Comp. in Simu., 2015, 108: 16–40.

    Article  MathSciNet  Google Scholar 

  28. Lux T, Herd behavior, bubbles and crashes, Econ. J., 1995, 105: 881–896.

    Article  Google Scholar 

  29. Gardini L, Merlone U, and Tramontana F, Inertia in binary choices: Continuity breaking and big-bang bifurcation points, J. of Econ. Behavior and Org., 2011, 80: 153–167.

    Article  Google Scholar 

  30. Puu T, The Hicksian trade cycle with floor and ceiling dependent on capital stock, J. Econ. Dyn. Control, 2007, 31: 575–592.

    Article  MathSciNet  Google Scholar 

  31. Gardini L, Avrutin V, and Sushko I, Codimension-2 border collision bifurcations in one-dimensional discontinuous piecewise smooth maps, Int. J. Bifur. Chaos, 2014, 24: 1450024-1-30

    Article  MathSciNet  Google Scholar 

  32. Gardini L and Tramontana F, Border collision bifurcation curves and their classification in a family of 1D discontinuous maps, Chaos, Solitons & Fractals, 2011, 44(4–5): 248–259.

    Article  MathSciNet  Google Scholar 

  33. Gardini L and Tramontana F, Border-collision bifurcations in 1D piecewise-linear maps and Leonovs approach, Int. J. of Bifur. and Chaos, 2010, 20(10): 3085–3104.

    Article  MathSciNet  Google Scholar 

  34. Gardini L and Tramontana F, Border collision bifurcations in 1D PWL map with one discontinuity use of the first return map, Int. J. of Bifur. and Chaos, 2010, 20(11): 3259–3574.

    Article  MathSciNet  Google Scholar 

  35. Sushko I, Gardini L, and Avrutin V, Nonsmooth one-dimensional maps: Some basic concepts and definitions, J. of Dif. Equ. and Appl., 2016, 22(12): 1816–1870.

    Article  MathSciNet  Google Scholar 

  36. Bischi G and Merlone U, Global dynamics in binary choice models with social influence, The J. of Math. Sociology, 2009, 33: 277–302.

    Article  Google Scholar 

  37. Bischi G, Gardini L, and Tramontana F, Bifurcation curves in discontinuous maps, Discrete and Continuous Dyn. Systems-Series B, 2012, 13(2): 249–267.

    Article  MathSciNet  Google Scholar 

  38. Gu E G, On the price dynamics of a two dimensional financial market model with entry levels, Complexity, 2020, Article ID: 3654083, 23 Pages.

  39. Gu E G and Guo J, BCB curves and contact bifurcations in piecewise linear discontinuous map arising in a financial market, Int. J. of Bifur. and Chaos, 2019, 29(2): 195002–195035.

    Article  MathSciNet  Google Scholar 

  40. Banerjee S, Ranjan P, and Grebogi C, Bifurcations in two dimensional piecewise smooth maps-theory and applications in switching circuits, IEEE Trans. Circuits. Syst. I, 2000, 47(5): 633–643.

    Article  Google Scholar 

  41. Chiarella C, Dieci R, and Gardini L, The dynamic interaction of speculation and diversification, Appl Math Finance, 2005, 12(1): 17–52.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Huanghe University of Science and Technology, Zhengzhou, 450006, China

    Enguo Gu

Authors
  1. Enguo Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Enguo Gu.

Ethics declarations

The authors declare no conflict of interest.

Additional information

This research was supported by the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities under Grant No. CZT20006.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, E. The Inherent Law of the Unpredictability of Financial Asset Price Fluctuations: Multistability and Chaos. J Syst Sci Complex 37, 776–804 (2024). https://doi.org/10.1007/s11424-024-1198-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-024-1198-4

Keywords

Search

Navigation