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Asymmetric COGARCH processes

Part of: Inference from stochastic processes Parametric inference Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Anita Behme ,
Claudia Klüppelberg  and
Kathrin Mayr
Anita Behme
Affiliation:
Center for Mathematical Sciences, Technische Universität München, 85748 Garching, Boltzmannstrasse 3, Germany. Email address: behme@ma.tum.de.
Claudia Klüppelberg
Affiliation:
Center for Mathematical Sciences, Technische Universität München, 85748 Garching, Boltzmannstrasse 3, Germany. Email address: cklu@ma.tum.de.
Kathrin Mayr
Affiliation:
Center for Mathematical Sciences, Technische Universität München, 85748 Garching, Boltzmannstrasse 3, Germany.
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Abstract

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Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH model. We calculate higher-order moments and extend the first-jump approximation. These results are prerequisites for moment estimation and pseudo maximum likelihood estimation of the GJR-COGARCH model parameters, respectively, which we derive in detail.

Keywords

APCOGARCHasymmetric power COGARCHCOGARCHfirst-jump approximationcontinuous-time GARCHGJR-GARCHGJR-COGARCHmaximum-likelihood estimationhigh-frequency datamethod of momentsstochastic volatility

MSC classification

Primary: 60G10: Stationary processes 60G51: Processes with independent increments; L'evy processes 62M05: Markov processes
Secondary: 62F10: Point estimation 62M10: Time series, auto-correlation, regression, etc.

Information

Type
Part 5. Finance and econometrics
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 161 - 173
Copyright
Copyright © Applied Probability Trust 2014 

References

Aït-Sahalia, Y., and Jacod, J. (2014). High-Frequency Financial Econometrics. Princeton University Press.Google Scholar
Behme, A., Lindner, A., and Maller, R. (2011). Stationary solutions of the stochastic differential equation dVt=Vt-dUt+dLt with Lévy noise. Stoch. Process. Appl. 121, 91–108.Google Scholar
Bibbona, E., and Negri, I. (2014). Higher moments and prediction based estimation for the {COGARCH}(1,1) model. Preprint. Available at http://arxiv.org/abs/1401.7819v2.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.CrossRefGoogle Scholar
Brockwell, P. J., and Davis, R. A. (1987). Time Series: Theory and Methods. Springer, New York.CrossRefGoogle Scholar
Ding, Z., Granger, C. W. J., and Engle, R. F. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1, 83–106.Google Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Fasen, V. (2010). Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes. Bernoulli 16, 51–79.Google Scholar
Franq, C., and Zakoian, J.-M. (2010). GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley, Chichester.Google Scholar
Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993). On the relation between expected return on stocks. J. Finance 48, 1779–1801.Google Scholar
Haug, S., Klüppelberg, C., Lindner, A., and Zapp, M. (2007). Method of moment estimation in the COGARCH(1,1) model. Econometrics J. 10, 320–341.Google Scholar
Hentschel, L. (1995). All in the family: nesting symmetric and asymmetric {GARCH} models. J. Financial Econom. 39, 71–104.Google Scholar
Jacod, J., and Protter, P. (2012). Discretization of Processes. Springer, Heidelberg.Google Scholar
Kim, M., and Lee, S. (2013). On the maximum likelihood estimator for irregularly observed time series data from COGARCH(1,1) models. REVSTAT Statist. J. 11, 135–168.Google Scholar
Klüppelberg, C., Lindner, A., and Maller, R. (2004). A continuous-time {GARCH} process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Prob. 41, 601–622.Google Scholar
Klüppelberg, C., Lindner, A., and Maller, R. (2006). Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance, eds Kabanov, Y., Liptser, R. and Stoyanov, J., Springer, Berlin, pp. 393–419.Google Scholar
Lee, O. (2010). A continuous time asymmetric power {GARCH} process driven by a Lévy process. J. Korean Data Inf. Sci. Soc. 21, 1311–1317.Google Scholar
Maller, R. A., Müller, G., and Szimayer, A. (2008). GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli 14, 519–542.CrossRefGoogle Scholar
Mayr, K. (2013). Der asymmetrische COGARCH: seine definition, approximation und schätzung. , Technische Universität München. Available at http://mediatum.ub.tum.de/node?id=1156296.Google Scholar
McKenzie, M., and Mitchell, H. (2002). Generalized asymmetric power ARCH modelling of exchange rate volatility. Appl. Financial Econom. 12, 555–564.Google Scholar
Mikosch, T., and Straumann, D. (2006). Stable limits of martingale transforms with application to the estimation of {GARCH} parameters. Ann. Statist. 34, 493–522.Google Scholar
Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45, 7–38.Google Scholar
Penzer, J., Wang, M., and Yao, Q. (2009). Approximating volatilities by asymmetric power {GARCH} functions. Austral. N. Z. J. Statist. 51, 201–225.CrossRefGoogle Scholar
Rabemananjara, R., and Zakoian, J.-M. (1993). Threshold ARCH models and asymmetries in volatility. J. Appl. Econometrics 8, 31–49.CrossRefGoogle Scholar
Stelzer, R. (2009). First jump approximation of a Lévy-driven SDE and an application to multivariate ECOGARCH processes. Stoch. Process. Appl. 119, 1932–1951.Google Scholar
Straumann, D. (2005). Estimation in Conditionally Heteroscedastic Time Series Models. Springer, Berlin.Google Scholar
Zakoian, J.-M. (1994). Threshold heteroskedastic models. J. Econom. Dynamics Control 18, 931–955.Google Scholar