zbMATH — the first resource for mathematics

GARCH modelling in continuous time for irregularly spaced time series data. (English) Zbl 1155.62067
Summary: The discrete-time GARCH methodology which has had such a profound influence on the modelling of heteroscedasticity in time series is intuitively well motivated in capturing many ‘stylized facts’ concerning financial series, and is now almost routinely used in a wide range of situations, often including some where the data are not observed at equally spaced intervals of time. However, such data is more appropriately analyzed with a continuous-time model which preserves the essential features of the successful GARCH paradigm. One possible such extension is the diffusion limit of D. B. Nelson [J. Econ. 45, No. 1–2, 7–38 (1990; Zbl 0719.60089)], but this is problematic in that the discrete-time GARCH model and its continuous-time diffusion limit are not statistically equivalent.
As an alternative, C. Klüppelberg et al. [J. Appl. Probab. 41, No. 3, 601–622 (2004; Zbl 1068.62093)] introduced a continuous-time version of the GARCH (the ‘COGARCH’ process) which is constructed directly from a background driving Lévy process. The present paper shows how to fit this model to irregularly spaced time series data using discrete-time GARCH methodology, by approximating the COGARCH with an embedded sequence of discrete-time GARCH series which converges to the continuous-time model in a strong sense (in probability, in the Skorokhod metric), as the discrete approximating grid grows finer. This property is also especially useful in certain other applications, such as options pricing. The way is then open to using, for the COGARCH, similar statistical techniques to those already worked out for GARCH models and to illustrate this, an empirical investigation using stock index data is carried out.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G51 Processes with independent increments; Lévy processes
Full Text: DOI arXiv
[1] Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed form approximation approach. Econometrica 70 223-262. · Zbl 1104.62323 · doi:10.1111/1468-0262.00274
[2] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus . Cambridge Univ. Press. · Zbl 1073.60002
[3] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press. · Zbl 0861.60003
[4] Bollerslev, T. (1986). Generalised autoregressive conditional heteroskedasticity. J. Econometrics 31 307-327. · Zbl 0616.62119 · doi:10.1016/0304-4076(86)90063-1
[5] Corradi, V. (2000). Reconsidering the continuous time limit of the GARCH(1, 1) process. J. Econometrics 96 145-153. · Zbl 0974.60063 · doi:10.1016/S0304-4076(99)00053-6
[6] Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the United Kingdom inflation. Econometrica 50 987-1007. · Zbl 0491.62099 · doi:10.2307/1912773
[7] Gihman, I.I. and Skorokhod, A.V. (1975). The Theory of Stochastic Processes. II . New York: Springer. · Zbl 0305.60027
[8] Goldie, C.M. and Maller, R.A. (2000). Stability of perpetuities. Ann. Probab. 28 1195-1218. · Zbl 1023.60037 · doi:10.1214/aop/1019160331
[9] Haug, S., Klüppelberg, C., Lindner, A. and Zapp, M. (2007). Method of moments estimation in the COGARCH(1, 1) model. Econom. J. 10 320-341. · Zbl 1186.91231 · doi:10.1111/j.1368-423X.2007.00210.x
[10] Jacod, J. (2006). Parametric inference for discretely observed non-ergodic diffusions. Bernoulli 12 383-402. · Zbl 1100.62081 · doi:10.3150/bj/1151525127 · euclid:bj/1151525127
[11] Jorion, P. (2000). Modeling Time-Varying Risk . New York: McGraw-Hill.
[12] Kallsen, J. and Taqqu, M.S. (1998). Option pricing in ARCH-type models. Math. Finance 8 13-26. · Zbl 0911.90028 · doi:10.1111/1467-9965.00042
[13] Kallsen, J. and Vesenmayer, B. (2008). COGARCH as a continuous time limit of GARCH(1, 1). Stohastic Process. Appl. · Zbl 1172.62025
[14] Klüppelberg, C., Lindner, A. and Maller, R.A. (2004). A continuous time GARCH process driven by a Lévy process: Stationarity and second order behaviour. J. Appl. Probab. 41 601-622. · Zbl 1068.62093 · doi:10.1239/jap/1091543413
[15] Klüppelberg, C., Lindner, A. and Maller, R.A. (2006). Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift (Yu. Kabanov, R. Lipster and J. Stoyanov, eds.) 393-419. Berlin: Springer. · Zbl 1124.60053
[16] Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space D [0, \infty ). J. Appl. Probab. 10 109-121. · Zbl 0258.60008 · doi:10.2307/3212499
[17] Maller, R.A., Solomon, D. and Szimayer, A. (2006). A multinomial approximation for American option prices in Lévy process models. Math. Finance 16 613-633. · Zbl 1130.91027 · doi:10.1111/j.1467-9965.2006.00286.x
[18] Maller, R.A., Müller, G. and Szimayer, A. (2008). Ornstein-Uhlenbeck processes and extensions. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.-P. Kreiß and T. Mikosch, eds.). New York: Springer. · Zbl 1185.60036
[19] McNeil, A.J. and Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. J. Emp. Fin. 7 271-300.
[20] Müller, G. (2007). MCMC estimation of the COGARCH(1, 1) model. Preprint, Munich Univ. Technology.
[21] Nelson, D.B. (1990). ARCH models as diffusion approximations. J. Econometrics 45 7-38. · Zbl 0719.60089 · doi:10.1016/0304-4076(90)90092-8
[22] Protter, P. (2005). Stochastic Integration and Differential Equations , 2nd ed. Heidelberg: Springer.
[23] Ritchken, P. and Trevor, R. (1999). Pricing options under GARCH and stochastic volatility processes. J. Finance 54 377-402.
[24] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press. · Zbl 0973.60001
[25] Szimayer, A. and Maller, R.A. (2007). Finite approximation schemes for Lévy processes, and their application to optimal stopping problems. Stochastic Process. Appl. 117 1422-1447. · Zbl 1125.60041 · doi:10.1016/j.spa.2007.01.012
[26] Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions. Ann. Statist. 30 754-783. · Zbl 1029.62006 · doi:10.1214/aos/1028674841 · euclid:aos/1028674841
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.