Stephan Haug
Exponential COGARCH and
other continuous time models
with applications to high frequency data.
December 2006
Center for Mathematical Sciences
Munich University of Technology
85747 Garching bei Mu¨nchenTechnische Universit¨at Mu¨nchen
Zentrum Mathematik
Lehrstuhl fu¨r Mathematische Statistik
Exponential COGARCH and other
continuous time models
with applications to high frequency data
Stephan Haug
Vollst¨andiger Abdruck der von der Fakult¨at fu¨r Mathematik der Technischen Uni-
versit¨at Mu¨nchen zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr.rer.nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. Jan Kallsen
Pru¨fer der Dissertation: 1. Univ.-Prof. Claudia Czado, Ph.D.
2. Prof. Peter Brockwell, Ph.D.,
Colorado State University, Fort Collins, USA
Die Dissertation wurde am 07.12.2006 bei der Technischen Universit¨at eingereicht
und durch die Fakult¨at fu¨r Mathematik am 23.03.2007 angenommen.Abstract: We address several approaches to modelling high frequency finan-
cial data in continuous time. Besides considering estimation for the existing con-
tinuous time GARCH(1,1) (COGARCH) process we will propose three new mod-
els. At first we suggest a method of moment estimator for the parameters of the
COGARCH(1,1) process. We show that the resulting estimators are consistent and
asymptotically normal and investigate the empirical quality in a simulation study
based on the compound Poisson and Variance Gamma driven COGARCH(1,1)
model. The model is also fitted to high-frequency financial data from the New York
Stock Exchange. In the following chapter we develop the first new model, an expo-
nentialCOGARCH(p,q)process.Weinvestigatestationarityandmomentproperties
of the model. An instantaneous leverage effect can beshown ifp=q. Thefirst steps
in estimating this new model are taken by proposing a quasi-maximum likelihood
type estimator for the parameters of a compound Poisson ECOGARCH(1,1) pro-
cess. To account for thestrongpersistence involatility, which is sometimes observed
in empiricaldata, wedevelop afractionally integrated ECOGARCH(p,d,q) process.
Similarly to the short memory case we investigate stationarity and moment prop-
erties of the model. It is also shown that the long memory effect introduced in the
log-volatility propagates to the volatility process. Finally considering absolute log
returns as a proxy for stochastic volatility, the influence of explanatory variables on
absolutelogreturnsofultrahighfrequencydataisanalysed.Inparticularwepropose
a new mixed effect model class for the absolute log returns. Explanatory variable
information is used to model the fixed effects, whereas the error is decomposed in a
non-negative L´evy driven continuous time ARMA process and a market microstruc-
ture noise component. Theparameters are estimated in a state space approach with
application of the Kalman filter. In a small simulation study the performance of the
estimators isinvestigated. ThemodelisappliedtoIBMtradedataandtheinfluence
of bid-ask spread and duration is quantified on a daily basis.To Nici
for sharing
so many
office hours
with me.Contents
Introduction 1
1 Preliminaries 6
1.1 L´evy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Stochastic integration . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.2 Itˆo’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 State space models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 Discrete time state space model . . . . . . . . . . . . . . . . . 24
1.3.2 Kalman filter and Gaussian likelihood . . . . . . . . . . . . . 25
1.4 Continuous time autoregressive moving average processes . . . . . . 27
1.4.1 L´evy driven CARMA(p,q) processes . . . . . . . . . . . . . . 27
1.4.2 L´evy driven FICARMA(p,d,q) processes . . . . . . . . . . . 32
1.5 Continuous time GARCH(1,1) processes . . . . . . . . . . . . . . . . 34
2 Estimating the COGARCH(1,1) 42
2.1 Identifiability of the model parameters . . . . . . . . . . . . . . . . . 42
2.2 The estimation algorithm . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Asymptotic properties of the moment estimators . . . . . . . . . . . 46
2.4 Examples of COGARCH(1,1) processes . . . . . . . . . . . . . . . . 51
2.4.1 Compound Poisson COGARCH(1,1) . . . . . . . . . . . . . . 51
2.4.2 Variance Gamma COGARCH(1,1) . . . . . . . . . . . . . . . 54
2.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.1 Estimation results for the compound Poisson COGARCH(1,1) 56
2.5.2 Estimation results for the Variance Gamma COGARCH(1,1) 56
22.5.3 Estimation of the volatility σ . . . . . . . . . . . . . . . . . 58t
2.6 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Exponential continuous time GARCH process 64
3.1 The discrete time EGARCH process . . . . . . . . . . . . . . . . . . 65
3.2 Exponential COGARCH . . . . . . . . . . . . . . . . . . . . . . . . . 66
iii
3.3 Second order properties of the volatility process . . . . . . . . . . . . 75
3.4 Second order properties of the return process . . . . . . . . . . . . . 78
3.4.1 Moments and autocovariance function of the return process . 78
3.4.2 Leverage effect . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 QMLE of compound Poisson ECOGARCH(1,1) 84
4.1 Quasi MLE of compound Poisson ECOGARCH(1,1) . . . . . . . . . 84
4.2 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.1 Leverage case with Gaussian jump distribution . . . . . . . . 88
4.2.2 Leverage case with student-t jump distribution . . . . . . . . 90
4.2.3 Non-leverage case. . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1 One-step ahead prediction of the volatility . . . . . . . . . . . 93
4.3.2 Prediction interval for the log-price . . . . . . . . . . . . . . . 98
4.4 Analysis of General Motors stock prices . . . . . . . . . . . . . . . . 99
5 Fractionally integrated ECOGARCH process 104
5.1 Fractionally integrated exponential COGARCH . . . . . . . . . . . . 105
5.2 Second order properties of the volatility process . . . . . . . . . . . . 110
5.3 Second order properties of the return process . . . . . . . . . . . . . 116
6 Mixedeffectmodelsforabsolutelogreturnsofultrahighfrequency
data 118
6.1 A mixed effect regression model for irregularly spaced data . . . . . 120
6.1.1 Regression mean specification . . . . . . . . . . . . . . . . . . 120
6.1.2 Correlated residuals . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.3 A generalised regression model with CARMA(p,q) random
effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Direct approach . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.2 State space approach . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Explorative data analysis . . . . . . . . . . . . . . . . . . . . 130
6.4.2 Estimation results . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.3 Analysis of the correlation structure . . . . . . . . . . . . . . 132
6.4.4 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4.5 Prediction results . . . . . . . . . . . . . . . . . . . . . . . . . 135
Conclusion and outlook 139