Application of Iterated Filtering for Parametric Estimation of Instantaneous Variance in the Case of Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Processes

Piotr Szczepocki

doi:10.5604/01.3001.0013.8364

Piotr Szczepocki Uniwersytet Łódzki, Wydział Ekonomiczno-Socjologiczny, Katedra Metod Statystycznych ORCID:https://orcid.org/0000-0001-8377-3831 Przegląd Statystyczny. Statistical Review, vol. 66, 2019, 1, pages: 51-68 Published online: 28 May 2019 DOI 10.5604/01.3001.0013.8364

1087 Views 65 Downloads

ARTICLE

(Polish) PDF

ABSTRACT

The article presents a method for parametric estimation of instantaneous variance in the case of non-Gaussian Ornstein-Uhlenbeck stochastic volatility process by means of the iterated filtering and realized variance estimator. The method is applied to realized variance of S&P500 index data. Empirical application is accompanied with simulation study to examine performance of the estimation technique.

KEYWORDS

stochastic volatility, Ornstein-Uhlenbeck process, iterated filtering

JEL

C51, C58

REFERENCES

Andersen T. G., Bollerslev T., Diebold F. X., Ebens H., (2001), The Distribution of Realised Stock Return Volatility, Journal of Financial Economics, 61, 43–76.

Andersen T. G., Bollerslev T., Diebold F. X., Labys P., (2001), The Distribution of Exchange Rate Volatility, Journal of the American Statistical Association, 96, 42–55.

Andrieu C., Doucet A., Holenstein R., (2010), Particle Markov Chain Monte Carlo Methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72 (3), 269–342.

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), 63 (2), 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), 64, (2), 253–280.

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.

Brzozowska-Rup K., Dawidowicz A. L., (2009), Metoda filtru cząsteczkowego, Matematyka stosowana: matematyka dla społeczeństwa, 10 (51), 69–107.

Cont R., Tankov P., (2003), Financial Modelling with Jump Processes, CRC Financial Mathematics Series, Chapman and Hall.

Del Moral P., (1996), Non-Linear Filtering: Interacting Particle Resolution, Markov Processes and Related Fields, 2 (4), 555–581.

Del Moral P., Doucet A., Jasra A., (2006), Sequential Monte Carlo for Bayesian Computation. Bayesian Statistics, 8, Oxford University Press, Oxford, UK.

Doman M., Doman R., (2009), Modelowanie zmienności i ryzyka: metody ekonometrii finansowej, Wolters Kluwer.

Doucet A., de Freitas N., Gordon N. J., (2001), Sequential Monte Carlo Methods in Practice, Springer, NewYork, NY.

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, 61 (1), 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, 137 (10), 3068–3081.

Gander M. P. S., Stephens D. A., (2007b), Simulation and Inference for Stochastic Volatility Models Driven by Lévy Processes, Biometrika, 94 (3),627–646.

Gerd H., Lunde A., Shephard N., Sheppard K., (2009), Oxford-Man Institute's Realized Library, Oxford-Man Institute, University of Oxford, dostęp online (8.04.2019): https://realized.oxfordman. ox.ac.uk.

Gordon N. J., Salmond D. J., Smith A. F., (1993), Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation, IEE Proceedings F (Radar and Signal Processing),140 (2), 107–113.

Griffin J. E., Steel M. F. J., (2006), Inference with Non-Gaussian Ornstein-Uhlenbeck Processes for Stochastic Volatility, Journal of Econometrics, 134, (2), 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, 54, (11), 2594–2608.

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, 11 (6), 917– –932.

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, 235 (11), 3355–3365.

James L. F., Kim D., Zhang Z., (2013), Exact Simulation Pricing with Gamma Processes and Their Extensions, arXiv preprint arXiv:1310.6526.

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, 36 (1), 75–87.

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 (3), 328–351.

King A. A., Nguyen D., Ionides E. L., (2016), Statistical Inference for Partially Observed Markov Processes via the R Package Pomp, Journal of Statistical Software, 69 (12), 1–43.

Kitagawa G., (1996), Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models, Journal of Computational and Graphical Statistics, 5 (1), 1–25.

Kliber P., (2013), Zastosowanie procesów dyfuzji ze skokami do modelowania polskiego rynku finansowego, Wydawnictwo Uniwersytetu Ekonomicznego w Poznaniu.

Iacus S. M., (2009), Simulation and Inference for Stochastic Differential Equations with R Examples, Springer Science & Business Media.

Ionides E. L., Bhadra A., Atchadé Y., King A., (2011), Iterated Filtering, Annals of Statistics, 39 (3), 1776–1802.

Ionides E. L., Bretó C., King A. A., (2006), Inference for Nonlinear Dynamical Systems, Proceedings of the National Academy of Sciences, 103 (49), 18438–18443.

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, 112 (3), 719–724.

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, 10 (7), 551–563.

Lindström E., Ionides E., Frydendall J., Madsen H., (2012), Efficient Iterated Filtering, IFAC Proceedings Volumes, 45 (16), 1785–1790.

Nguyen D., (2016), Another Look At Bayes Map Iterated Filtering, Statistics & Probability Letters, 118, 32–36.

Nicolato E., Venardos E., (2003), Option Pricing in Stochastic Volatility Models of the Ornstein- Uhlenbeck Type, Mathematical Finance, 13 (4), 445–466.

Pitt M. K., Shephard N., (1999), Filtering Via Simulation: Auxiliary Particle Filters, Journal of the American statistical association, 94 (446), 590–599.

Raknerud A., Skare O., (2012), Indirect Inference Methods for Stochastic Volatility Models Based on Non-Gaussian Ornstein-Uhlenbeck Processes, Computational Statistics & Data Analysis, 56 (11), 3260–3275.

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), 66 (2),369–393.

Schoutens W., (2003), Lévy Process in Finance. Pricing Financial Derivatives, John Wiley & Sons Ltd.

Taufer E., Leonenko N., Bee M., (2011), Characteristic Function Estimation of Ornstein–Uhlenbeck- Based Stochastic Volatility Models, Computational Statistics & Data Analysis, 55 (8), 2525– –2539.

Weron A., Weron R., (1998), Inżynieria finansowa, WNT, Warszawa.