Zastosowanie filtru Kalmana do modeli stochastycznej zmienności typu Ornsteina‑Uhlenbecka

Piotr Szczepocki

doi:10.18778/0208-6018.337.12

Authors

Piotr Szczepocki Uniwersytet Łódzki, Wydział Ekonomiczno‑Socjologiczny, Katedra Metod Statystycznych

DOI:

https://doi.org/10.18778/0208-6018.337.12

Keywords:

stochastic volatility models, Levy processes

Abstract

Barndorff‑Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility process is the Ornstein‑Uhlenbeck process driven by a Levy process without gaussian component. Parameter estimation of these models is difficult because the appropriate likelihood functions do not have a closed‑form expression. The article deals with application of the Kalman filter technique for parameter estimation of such models. The method is applied to EUR/PLN daily exchange rate data. Empirical application is accompanied with simulation study to examine statistical properties of the estimators.

Downloads

Download data is not yet available.

References

Andrieu C., Doucet A., Holenstein R. (2010), Particle Markov Chain Monte Carlo Methods, „Journal of the Royal Statistical Society: Series B (Statistical Methodology)”, t. 72(3), s. 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)”, t. 63, nr 2, s. 167–241. doi: 10.1111/1467–9868.00282.

Barndorff‑Nielsen O.E., Stelzer R. (2013), The multivariate supOU stochastic volatility model, „Mathematical Finance”, t. 23(2), s. 275–296.

Bertoin J. (1996), Lévy processes, t. 121, Cambridge Tracts in Mathematics, Cambridge University Press, London.

Bibby B.M., Sørensen M. (1995), Martingale estimation functions for discretely observed diffusion processes, „Bernoulli”, t. 1, nr 1/2, s. 17–39, doi: 10.2307/3318679.

Byrd R.H., Lu P., Nocedal J., Zhu C. (1995), A limited memory algorithm for bound constrained optimization, „SIAM J. Scientific Computing”, t. 16, nr 5, s. 1190–1208, doi: 10.1137/0916069.

Cont R., Tankov P. (2004), Financial Modelling with jump processes, Chapman & Hall/CRC, Boca Raton.

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”, t. 137, nr 10, s. 3068–3081, doi: 10.1016/j.jspi.2006.07.015.

Gander M.P.S., Stephens D.A. (2007b), Simulation and inference for stochastic volatility models driven by levy processes, „Biometrika”, t. 94, nr 3, s. 627–646, doi: 10.1093/biomet/asm048.

Gourieroux C., Monfort A., Renault E. (1993), Indirect inference, „Journal of Applied Econometrics”, t. 8, nr 1, s. S85–S118, doi: 10.1002/jae.3950080507.

Grewal M., Andrews A. (2010), Applications of Kalman filtering in aerospace 1960 to the present, „Historical perspectives. IEEE Control Systems Magazine”, t. 30, nr 3, s. 69–78, doi: 10.1109/mcs.2010.936465.

Griffin J.E., Steel M.F.J. (2006), Inference with non‑gaussian Ornstein‑Uhlenbeck processes for stochastic volatility, „Journal of Econometrics”, t. 134, nr 2, s. 605–644, doi: 10.1016/j.jeconom.2005.07.007.

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”, t. 54, nr 11, s. 2594–2608, doi: 10.1016/j.csda.2009.06.008.

Hamilton J.D. (1994), State‑space models, [w:] R.F. Engle, Handbook of econometrics, t. 4, North Holland, Amsterdam.

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”, t. 11(6), s. 917–932.

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

Nicolato E., Venardos E. (2003), Option pricing in stochastic volatility models of the Ornstein‑Uhlenbeck type, „Mathematical Finance”, t. 13, nr 4, s. 445–466, doi: 10.1111/1467–9965.t01–1–00023.

Parkinson M. (1980), The extreme value method for estimating the variance of the rate of return, „Journal of Business”, t. 53, nr 1, s. 61–65.

Pigorsch C., Stelzer R. (2009), A multivariate Ornstein‑Uhlenbeck type stochastic volatility model, https://mediatum.ub.tum.de/doc/1079183/file.pdf [dostęp: 28.01.2018].

Pitt M.K., Shephard N. (1999), Filtering via simulation: Auxiliary particle filters, „Journal of the American Statistical Association”, t. 94, nr 446, s. 590–599, doi: 10.2307/2670179.

Roberts G.O., 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)”, t. 66, nr 2, s. 369–393, doi: 10.1111/j.1369–7412.2004.05139.x.

Stelzer R., Tosstorff T., Wittlinger M. (2015), Moment based estimation of supOU processes and a related stochastic volatility model, „Statistics & Risk Modeling”, t. 32(1), s. 1–24.

Taufer E., Leonenko N. (2009), Simulation of Levy‑driven Ornstein‑Uhlenbeck processes with given marginal distribution, „Computational Statistics & Data Analysis”, t. 53(6), s. 2427–2437.

Taufer E., Leonenko N., Bee M. (2011), Characteristic function estimation of Ornstein‑Uhlenbeck‑based stochastic volatility models, „Computational Statistics & Data Analysis”, t. 55(8), s. 2525–2539.