Parameter Estimation in a Weak Hidden Markov Model with Independent Drift and Volatility

  • Xiaojing Xi
  • Rogemar S. MamonEmail author
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 209)


We develop a multivariate higher-order Markov model, also known as weak hidden Markov model (WHMM), for the evolution of asset prices. The means and volatilities of asset’s log-returns are governed by a second-order Markov chain in discrete time. WHMM enriches the usual HMM by incorporating more information from the past thereby capturing presence of memory in the underlying market state. A filtering technique in conjunction with the Expectation-Maximisation algorithm is adopted to develop the optimal estimates of model parameters. To ensure that the errors between the “true” parameters and estimated parameters are due only to the estimation method and not from model uncertainty, recursive filtering algorithms are implemented to a simulated dataset. Using goodness-of-fit metrics, we show that the WHMM-based filtering techniques are able to recover the “true” underlying parameters.


Markov Chain Asset Price Risky Asset GARCH Model Financial Time Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ang, A., Timmermann, A.: Regime changes and financial markets. Ann. Rev. Financ. Econ. 4, 313–337 (2012)CrossRefGoogle Scholar
  2. 2.
    Bulla, J., Bulla, I.: Stylised facts of financial time series and hidden semi-Markov models. Comput. Stat. Data Anal. 51, 2192–2209 (2006)CrossRefGoogle Scholar
  3. 3.
    Calvet, L.E., Fisher, A.J.: Multifrequency news and stock returns. J. Financ. Econ. 86, 178–212 (2007)CrossRefGoogle Scholar
  4. 4.
    Ching, W.K., Siu, T.K., Li, L.M.: Pricing exotic options under a higher-order Markovian regime. J. Appl. Math. Decis. Sci. 1–15 (2007) Article ID 18014Google Scholar
  5. 5.
    Elliott, R., Moore, J., Aggoun, L.: Hidden Markov Models: Estimation and Control. Springer, New York (1995)Google Scholar
  6. 6.
    Elliott, R.J., Malcolm, W.P., Tsoi, A.H.: Robust parameter estimation for asset price models with Markov modulated volatilities. J. Econ. Dyn. Control 27, 1391–1409 (2003)CrossRefGoogle Scholar
  7. 7.
    Elliott, R.J., Siu, T.K., Badescu. A.: On mean-variance portfolio selection under a hidden Markovian regime-switching model. Econ. Model. 27, 678–686 (2010)Google Scholar
  8. 8.
    Erlwein, C., Mamon, R.: An online estimation scheme for a Hull-White model with HMM-driven parameters. Stat. Methods Appl. 18(1), 87–107 (2009)CrossRefGoogle Scholar
  9. 9.
    Erlwein, C., Mamon, R., Davison, M.: An examination of HMM-based investment strategies for asset allocation. Appl. Stoch. Model Bus. Ind. 27, 204–221 (2011)CrossRefGoogle Scholar
  10. 10.
    Hamilton, J.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384 (1989)CrossRefGoogle Scholar
  11. 11.
    Hardy. M.R.: A regime-switching model of long-term stock returns. North Am. Actuar. J. 5, 41–53 (2001)Google Scholar
  12. 12.
    Hunt, J., Devolder, P.: Semi-Markov regime switching interest rate models and minimal entropy measure. Phys. A Stat. Mech. Appl. 390, 3767–3781 (2011)CrossRefGoogle Scholar
  13. 13.
    Lange, T., Rahbek, A.: An introduction to regime switching time series model. In: Lange, T., Rahbek, A. (eds.) Handbook of Financial Time Series, pp. 871–887. Springer, Berlin/Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Leroux. B.G.: Maximum-likelihood estimation for hidden Markov models. Stoch. Process. Appl. 40, 127–143 (1992)Google Scholar
  15. 15.
    Liew, C.C, Siu, T.K.: A hidden Markov-switching model for option valuation. Insur. Math. Econ. 47, 374–384 (2010)CrossRefGoogle Scholar
  16. 16.
    Luo, S., Tsoi, A.H.: Filtering of hidden weak Markov chain-discrete range observation. In: Mamon, R.S., Elliott, R.J. (eds.) Hidden Markov Models in Finance, pp. 106–119. Springer, New York (2007)Google Scholar
  17. 17.
    Mamon, R.S., Jalen. L.: Parameter estimation in a regime-switching model when the drift and volatility are independent. In: Proceedings of 5th International Conference on Dynamic Systems and Applications, pp. 291–298. Dynamic Publishers, Atlanta (2008)Google Scholar
  18. 18.
    Mamon, R.S., Erlwein, C., Gopaluni, R.B.: Adaptive signal processing of asset price dynamics with predictability analysis. Inform. Sci. 178, 203–219 (2008)CrossRefGoogle Scholar
  19. 19.
    Pagan, A.R., Schwert, G.W.: Alternative models for conditional stock volatility. J. Econ. 45, 267–290 (1990)CrossRefGoogle Scholar
  20. 20.
    Rydén, T., Terasvirta, T., Asbrink, S.: Stylized facts of daily return series and the hidden Markov model. J. Appl. Econ. 13, 217–244 (1998)CrossRefGoogle Scholar
  21. 21.
    Siu, T., Ching, W., Fung, E., Ng, M. Li, X.: A high-order Markov-switching model for risk measurement. Comput. Math. Appl. 58, 1–10 (2009)CrossRefGoogle Scholar
  22. 22.
    Tyssedal, J.S., Tjostheim, D.: An autoregressive model with suddenly changing parameters and an application to stock market prices. Appl. Stat. 37, 353–369 (1988)CrossRefGoogle Scholar
  23. 23.
    Veronesi, P.: How does information affect stock returns? J. Financ. 55, 807–837 (2009)CrossRefGoogle Scholar
  24. 24.
    Wu, C.: On the convergence properties of the EM algorithm. Ann. Stat. 11, 95–103 (1983)CrossRefGoogle Scholar
  25. 25.
    Xi, X., Mamon, R.S.: Parameter estimation of an asset price model driven by a weak hidden Markov chain. Econ. Model. 28, 36–46 (2011)CrossRefGoogle Scholar
  26. 26.
    Yu, S., Liu, Z., Squillante, M.S., Xia, C.H., Zhang, L.: A hidden semi-Markov model for web workload self-similarity. In: Proceedings of 21st IEEE International Performance, Computing, and Communications Conference, pp. 65–72, Phoenix (2002)Google Scholar
  27. 27.
    Zhou, N., Mamon, R.: An accessible implementation of interest rate models with Markov-switching. Expert Syst. Appl. 39(5), 4679–4689 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsLondonCanada
  2. 2.Department of Statistical & Actuarial SciencesUniversity of Western OntarioLondonCanada

Personalised recommendations