• Bernoulli
  • Volume 14, Number 2 (2008), 519-542.

GARCH modelling in continuous time for irregularly spaced time series data

Ross A. Maller, Gernot Müller, and Alex Szimayer

Full-text: Open access


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 Nelson, but this is problematic in that the discrete-time GARCH model and its continuous-time diffusion limit are not statistically equivalent. As an alternative, Klüppelberg et al. recently 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.

Article information

Bernoulli, Volume 14, Number 2 (2008), 519-542.

First available in Project Euclid: 22 April 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

COGARCH process continuous-time GARCH process Lévy process pseudo-maximum likelihood estimation Skorokhod distance stochastic volatility


Maller, Ross A.; Müller, Gernot; Szimayer, Alex. GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli 14 (2008), no. 2, 519--542. doi:10.3150/07-BEJ6189.

Export citation


  • [1] Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed form approximation approach., Econometrica 70 223–262.
  • [2] Applebaum, D. (2004)., Lévy Processes and Stochastic Calculus. Cambridge Univ. Press.
  • [3] Bertoin, J. (1996)., Lévy Processes. Cambridge Univ. Press.
  • [4] Bollerslev, T. (1986). Generalised autoregressive conditional heteroskedasticity., J. Econometrics 31 307–327.
  • [5] Corradi, V. (2000). Reconsidering the continuous time limit of the GARCH(1, 1) process., J. Econometrics 96 145–153.
  • [6] Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the United Kingdom inflation., Econometrica 50 987–1007.
  • [7] Gihman, I.I. and Skorokhod, A.V. (1975)., The Theory of Stochastic Processes. II. New York: Springer.
  • [8] Goldie, C.M. and Maller, R.A. (2000). Stability of perpetuities., Ann. Probab. 28 1195–1218.
  • [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.
  • [10] Jacod, J. (2006). Parametric inference for discretely observed non-ergodic diffusions., Bernoulli 12 383–402.
  • [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.
  • [13] Kallsen, J. and Vesenmayer, B. (2008). COGARCH as a continuous time limit of GARCH(1, 1)., Stohastic Process. Appl. To appear.
  • [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.
  • [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.
  • [16] Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space, D[0, ∞). J. Appl. Probab. 10 109–121.
  • [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.
  • [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. To appear.
  • [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.
  • [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.
  • [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.
  • [26] Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions., Ann. Statist. 30 754–783.