Estimating parameters of a stochastic volatility model using the expectation-maximization algorithm coupled with a Gaussian particle filter

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Tanit Malakorn
Thanapat Iamtan


In this paper, the expectation-maximization algorithm coupled with a Gaussian particle filter for maximum likelihood parameter estimation of a stochastic volatility model is investigated. Two data sets are provided for demonstration purposes: simulated data and daily foreign exchange rates data. Simulation studies illustrate that the parameter estimate trajectories are likely to converge to the true ones. When comparing the empirical results obtained from the conventional method and the proposed method, it can be seen that the parameter estimates from both methods are consistent with each other; however, the computational time is considerably reduced when using the method presented here.


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[1] Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654.

[2] Taylor, S.J., 1982. Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices 1961–79. In: Anderson, O.D. (ed). Time series analysis: theory and practice, 203–226.

[3] Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodology) 39, 1–38.

[4] Simon, D., 2010. Kalman filtering with state constraints: a survey of linear and nonlinear algorithms. IET Control Theory & Application 4, 1303–1318.

[5] Crassidis, J.L., Markley, F.L., Cheng, Y., 2007. Survey of nonlinear attitude estimation methods. Journal of Guidance, Control, and Dynamic 30, 12–28.

[6] Voss, H.U., Timmer, J., Kurths, J., 2004. Nonlinear dynamical system identification from uncertain and indirect measurements. International Journal of Bifurcation and Chaos 14, 1905–1933.

[7] Doucet, A., de Freitas, N., Gordon, N., 2001. Sequential Monte Carlo Methods in Practice, 1st ed. Springer Verlag, New York

[8] Gordon, N.J., Salmond, D.J., Smith, A.F.M., 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F - Radar and Signal Processing 140, 107–113.

[9] Kotecha, J.H., Djuric, P.M., 2003. Gaussian particle filtering. IEEE Transactions on Signal Processing 51, 2592–2601.

[10] Platanioti, K., McCoy, E.J., Stephens, D.A., 2005. A review of stochastic volatility: univariate and multivariate models, Imperial College, London

[11] McLachlan, G.L., Krishnan, T., 2008. The EM Algorithm and Extensions, 2nd ed. John Wiley & Sons, New York

[12] Brockwell, P.J., Davis, R.A., 2016. Introduction to Time Series and Forecasting, 3rd ed. Springer-Verlag, New York

[13] Commandeur, J.J.F., Koopman, S.J., 2007. An Introduction to State Space Time Series Analysis, Oxford University Press, Oxford

[14] Zia, A., Kirubarajan, T., Reilly, J.P., Yee, D., Punithakumar, K., Shirani, S., 2008. An EM algorithm for nonlinear state estimation with model uncertainties. IEEE Transactions on Signal Processing 56, 921–936.

[15] Godsill, S.J., Doucet, A., West, M., 2004. Monte Carlo smoothing for nonlinear time series. Journal of the American Statistical Association 99, 156–168.

[16] Sarkka, S., 2013. Bayesian filtering and smoothing, 1st ed. Cambridge University Press, Cambridge

[17] Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T., 2002. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing 50, 174–188.

[18] Cappe, O., Moulines, E., Ryden, T., 2005. Inference in Hidden Markov Models, 1st ed. Springer-Verlag, New York

[19] Kantas, N., Doucet, A., Singh, S.S., Maciejowski, J., Chopin, N., 2015. On particle methods for parameter estimation in state-space models. Statistical Science 30, 328–351.

[20] Xu, W., 2016. Estimation of dynamic panel data models with stochastic volatility using particle filters. Econometrics 4, 1–13.

[21] Ruiz, H.-C., Kappen, H.J., 2017. Particle smoothing for hidden diffusion processes: adaptive path integral smoother. IEEE Transactions on Signal Processing 65, 3191–3203.

[22] Finke, A., Singh, S.S., 2017. Approximate smoothing and parameter estimation in high-dimensional state-space models. IEEE Transactions on Signal Processing 65, 5982–5994.

[23] Wang, H., Nagayama, T., Su, D., 2017. Vehicle parameter identification through particle filter using bridge responses and estimated profile. Procedia Engineering 188, 64–71.

[24] Murakami, A., Shinmura, H., Ohno, S., Fujisawa, K., 2017. Model identification and parameter estimation of elastoplastic constitutive model by data assimilation using the particle filter. International Journal for Numerical and Analytical Methods in Geomechanics 42, 110–131.