Skip to main content

A Stochastic Chartist–Fundamentalist Model with Time Delays


A stochastic chartist–fundamentalist model of speculative asset dynamics in financial markets is developed. The model is represented by a stochastic delay-differential equation (SDDE). The SDDE is then solved using approximation and numerical Monte Carlo methods. The results show that for large time delays, the SDDE generates market-like stock price dynamics that reflect the memory effects of the time delay. The resultant dynamics agree with the empirical observation of the tendency of stock markets to deviate from pure random walk.

This is a preview of subscription content, access via your institution.


  1. Anufriev, M. (2007, April). Heterogeneous beliefs under different market architectures. Paper presented at the Workshop on Statistical Physics and Financial Markets, Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.

  2. Bukwar E. (2000) Introduction to the numerical analysis of stochastic delay differential equations. Journal of Computational and Applied Mathematics 125: 297–307

    Article  Google Scholar 

  3. Chiarella, C. (2007, April). The stochastic price dynamics of speculative behavior. Paper presented at the Workshop on Statistical Physics and Financial Markets, Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.

  4. Chiarella C., Dieci R., Gardini L. (2002) Speculative behavior and complex asset price dynamics: A global analysis. Journal of Economic Behavior and Organization, 49(2): 173–197

    Google Scholar 

  5. Dibeh G. (2005) Speculative dynamics in a time delay model of asset prices. Physica A 355(1): 199–208

    Article  Google Scholar 

  6. Dibeh G. (2007) Contagion effects in a chartist–fundamentalist model with time delays. Physica A 382(1): 52–57

    Article  Google Scholar 

  7. Frank T. D. (2006) Time-dependent solutions for stochastic systems with delays: Perturbation theory and applications to financial physics. Physics Letters A 357: 275–283

    Article  Google Scholar 

  8. Franke R., Sethi R. (1998) Cautious trend-seeking and complex asset price dynamics. Research in Economics 52(1): 61–79

    Article  Google Scholar 

  9. Guillouzic S., L’Hereux I., Longtin A. (1999) Small delay approximation of stochastic delay differential equations. Physical Review E 59(4): 3970–3982

    Article  Google Scholar 

  10. Guillouzic S., L’Hereux I., Longtin A. (2000) Rate processes in a delayed, stochastically driven, and overdamped system. Physical Review E 61(5): 4906–4914

    Article  Google Scholar 

  11. Johnson H., Shanno D. (1987) Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22(2): 143–152

    Article  Google Scholar 

  12. Johnson N., Jefferies P., Hu P. M. (2002) Financial Market Complexity. Oxford University Press, Oxford

    Google Scholar 

  13. Kuchler U., Platen E. (2002) Weak discrete time approximation of stochastic differential equations with time delay. Mathematics and Computers in Simulation 59: 497–505

    Article  Google Scholar 

  14. Lux T. (1995) Herd behavior, bubbles and crashes. Economic Journal 105: 881–896

    Article  Google Scholar 

  15. Ohanian, L.E. (1996). When the bubble bursts: Psychology or Fundamentals? Business Review of the Federal Reserve Bank of Philadelphia (Jan–Feb), 3–13.

  16. Pikovsky A., Kurths J. (1997) Coherence resonance in a noise-driven excitable system. Physical Review Letters 78: 775–778

    Article  Google Scholar 

  17. Rumelin W. (1982) Numerical treatment of stochastic differential equations. SIAM Journal of Numerical Analysis 19: 604–613

    Article  Google Scholar 

  18. Sethi R (1996) Endogenous Regime Switching in Speculative Markets. Structural Change and Economic Dynamics 7((1): 99–118

    Article  Google Scholar 

  19. Tambe S. S., Inamdar S. R., Kulkarni B. D. (1995) Diffusive broadening of limit cycle in presence of noise: A case study of reversible Brusselator. Mathematcis and Computers in Simulation 39: 115–124

    Article  Google Scholar 

  20. Westerhoff F. (2003) Speculative markets and the effectiveness of price limits. Journal of Economic Dynamics and Control 28: 493–508

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Haidar M. Harmanani.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dibeh, G., Harmanani, H.M. A Stochastic Chartist–Fundamentalist Model with Time Delays. Comput Econ 40, 105–113 (2012).

Download citation


  • Speculative models
  • Stochastic models
  • Delay-differential equations