Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

Ornstein–Uhlenbeck process, stochastic volatility, iterated filtering

ALTMAN, E. I., (1968). Financial Ratios, Discriminant Analysis and the Prediction of the Corporate Bankruptcy, The Journal of Finance, Vol. 23, pp. 589–609.

ANDRIEU, C., DOUCET, A., HOLENSTEIN, R., (2010). Particle Markov Chain Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 72, No. 3, pp. 269–342.

BHADRA, A., IONIDES, E. L., LANERI, K., PASCUAL, M., BOUMA, M., DHIMAN, R. C., (2011). Malaria in Northwest India: Data analysis via partially observed stochastic differential equation models driven by Lévy noise, Journal of the American Statistical Association, Vol. 106, No. 494, pp. 440–451.

BARNDORFF-NIELSEN, O.E. SHEPHARD, N., (2001). Non-Gaussian Ornstein- Uhlenbeck-based models and some of their uses in financial economics, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 63, No. 2, pp. 167–241.

BARNDORFF-NIELSEN, O. E., SHEPHARD, N., (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 64, No. 2, pp. 253–280.

BARNDORFF-NIELSEN, O. E., SHEPHARD, N., (2003). Integrated OU processes and Non-Gaussian OU-based stochastic volatility models, Scandinavian Journal of Statistics, Vol. 30, No. 2, pp. 277–295.

BENTH, F. E., GROTH, M., KUFAKUNESU, R., (2007). Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein–Uhlenbeck Stochastic Volatility Model, Applied Mathematical Finance, 14(4), 347–363.

BENTH, F. E., KARLSEN, K. H., REIKVAM, K., (2003). Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, Vol. 13, No. 2, pp. 215– 244.

BENTH, F. E., KARLSEN, K. H., (2005). A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets, Stochastics an International Journal of probability and Stochastic Processes, Vol. 77, No. 2, pp. 109– 137

BENTH, F. E., MEYER-BRANDIS, T., (2005). The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps, Finance and Stochastics, Vol. 9, No. 4, pp. 563–575.

BRETÓ, C., (2014). On idiosyncratic stochasticity of financial leverage effects, Statistics & Probability Letters, Vol. 91, pp. 20–26.

CAPPÉ, O., MOULINES, E., RYDÉN, T., (2009). Inference in Hidden Markov Models, Springer Series in Statistics, Springer.

CHOPIN, N., JACOB, P. E., PAPASPILIOPOULOS, O., (2013). SMC2: an efficient algorithm for sequential analysis of state space models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 75, No. 3, pp. 397–426.

CREAL, D. D., (2008). Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models, Computational Statistics & Data Analysis, Vol. 52, No. 6, pp. 2863–2876.

DOUCET, A., JACOB, P. E., RUBENTHALER, S., (2013). Derivative-free estimation of the score vector and observed information matrix with application to state-space models, arXiv preprint, URL: https://arXiv:1304.5768 .

DURBIN, J., KOOPMAN, S. J., (2012). Time series analysis by state space methods (Vol. 38), Oxford University Press.

FRÜHWIRTH-SCHNATTER, S., SÖGNER, L., (2009). Bayesian estimation of stochastic volatility models based on OU processes with marginal Gamma law, Annals of the Institute of Statistical Mathematics, Vol. 61, No. 1, pp. 159–179.

GANDER, M. P. S., STEPHENS, D. A., (2007a). Stochastic volatility modelling in continuous time with general marginal distributions: Inference, prediction and model selection, Journal of Statistical Planning and Inference, Vol. 137, No. 10, pp. 3068–3081.

GANDER, M. P. S., STEPHENS, D. A., (2007b). Simulation and inference for stochastic volatility models driven by levy processes, Biometrika, Vol. 94, No. 3, pp. 627–646.

GORDON, N. J., SALMOND, D. J., SMITH, A. F., (1993, April). Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings F-radar and signal processing, Vol. 140, No. 2, pp. 107–113.

GOURIEROUX, C., MONFORT A., RENAULT E., (1993). Indirect inference, Journal of Applied Econometrics, Vol. 8, pp. 85–118.

GRIFFIN, J. E., STEEL, M. F. J., (2006), Inference with non-Gaussian Ornstein– Uhlenbeck processes for stochastic volatility, Journal of Econometrics, Vol. 134, No. 2, pp. 605–644.

GRIFFIN, J. E., STEEL, M .F. J., (2010). Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein–Uhlenbeck processes, Computational Statistics & Data Analysis, Vol. 54, No. 11, pp. 2594– 2608.

HE, D., IONIDES, E. L., KING, A. A., (2009). Plug-and-play inference for disease dynamics: measles in large and small populations as a case study, Journal of the Royal Society Interface, Vol. 7, No. 43, pp. 271–283.

HUBALEK, F., POSEDEL, P., (2006). Asymptotic analysis for an optimal estimating function for Barndorff-Nielsen-Shephard stochastic volatility models, Work in progress, URL: https://arxiv.org/abs/0807.3479 .

HUBALEK, F., POSEDEL, P., (2011). Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models, Quantitative Finance, Vol. 11, No. 6, pp. 917–932.

HUBALEK, F., SGARRA, C., (2009). On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps, Stochastic Processes and their Applications, Vol. 119, No. 7, pp. 2137–2157.

HUBALEK, F., SGARRA, C., (2011). On the explicit evaluation of the geometric Asian options in stochastic volatility models with jumps, Journal of Computational and Applied Mathematics, Vol. 235, No. 11, pp. 3355–3365.

IONIDES, E. L., BRETÓ, C., KING, A. A., (2006). Inference for nonlinear dynamical systems, Proceedings of the National Academy of Sciences, Vol. 103, No. 49, pp. 18438–18443.

IONIDES, E. L., BHADRA, A., ATCHADÉ, Y., KING, A., (2011). Iterated filtering, The Annals of Statistics, Vol. 39, No. 3, pp. 1776–1802.

IONIDES, E. L., NGUYEN, D., ATCHADÉ, Y., STOEV, S., KING, A. A., (2015). Inference for dynamic and latent variable models via iterated, perturbed Bayes maps, Proceedings of the National Academy of Sciences, Vol. 112, No. 3, pp. 719– 724.

JAMES, L. F., MÜLLER, G., ZHANG, Z., (2018). Stochastic Volatility Models based on OU-Gamma time change: Theory and estimation, Journal of Business & Economic Statistics, Vol. 36, No. 1, pp. 75-87.

KING, A. A., IONIDES, E. L., PASCUAL, M., BOUMA, M. J., (2008). Inapparent infections and cholera dynamics, Nature, Vol. 454, No. 7206, pp. 877–880.

KING, A. A., IONIDES, E. L., BRETÓ, C., ELLNER, S., KENDALL, B., WEARING, H., FERRARI, M. J., LAVINE, M., REUMAN, D. C., (2010). POMP: statistical inference for partially observed Markov processes (R package), URL http://pomp.rforge. r-rproject.org.

LELE, S. R., DENNIS, B., LUTSCHER, F., (2007). Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods, Ecology letters, Vol. 10, No. 7, pp. 551–563.

NICOLATO, E. VENARDOS, E., (2003). Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type, Mathematical Finance, Vol. 13, No. 4, pp. 445– 466.

NGUYEN, D., (2016). Another look at Bayes map iterated filtering, Statistics & Probability Letters, Vol. 118, pp. 32–36.

R DEVELOPMENT CORE TEAM, (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, URL http://www.R-project.org.

ROBERTS, G., PAPASPILIOPOULOS, O., DELLAPORTAS, P., (2004). Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 66, No. 2, pp. 369–393.

RAKNERUD, A., SKARE, O., (2012). Indirect inference methods for stochastic volatility models based on non-Gaussian Ornstein–Uhlenbeck processes, Computational Statistics & Data Analysis, Vol. 56, No. 11, pp. 3260–3275.

SORENSEN, M., (2000). Prediction-based estimating functions. The Econometrics Journal, Vol. 3, No. 2, pp. 123–147.

SCHOUTENS, W., (2003). Lévy processes in finance. Wiley.

SZCZEPOCKI, P., (2018). Application of Kalman Filter to Stochastic Volatility Models of the Orstein Uhlenbeck Type, Acta Universitatis Lodziensis. Folia Oeconomica, Vol. 337, No. 4, pp. 183–201.

TAUFER, E., LEONENKO, N., BEE, M., (2011). Characteristic function estimation of Ornstein–Uhlenbeck-based stochastic volatility models, Computational Statistics & Data Analysis, Vol. 55, No. 8, pp. 2525–2539.

TONI, T., WELCH, D., STRELKOWA, N., IPSEN, A., STUMPF, M. P., (2008). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems, Journal of the Royal Society Interface, Vol. 6, No. 31, pp. 187–202.

Copyright © 2019 Statistics Poland