A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

Klüppelberg, Claudia; Lindner, Alexander; Maller, Ross

doi:10.1017/s0021900200020428

Cited by 86 publications

(156 citation statements)

References 16 publications

Supporting

Mentioning

150

Contrasting

Unclassified

Order By: Relevance

“…The COGARCH(p, q) process generalizes the COGARCH(1, 1) process that has been introduced following different arguments from those for the (p, q) case. However, choosing q = 1 and p = 1 in (1), the COGARCH(1, 1) process developed in Klüppelberg et al (2004) and Haug et al (2007) can be retrieved through straightforward manipulations and, for obtaining the same parametrization as in Proposition 3.2 of Klüppelberg et al (2004), the following equalities are necessary:…”

Section: Cogarch(p Q) Models Driven By a Lévy Processmentioning

confidence: 99%

“…As shown in Klüppelberg et al (2004) and remarked in Brockwell et al (2006), the condition in (6) is also necessary for the COGARCH(1, 1) case and can be simplified as:…”

Section: Cogarch(p Q) Models Driven By a Lévy Processmentioning

confidence: 99%

“…The continuous-time GARCH(1, 1) process has been introduced in Klüppelberg, Lindner, and Maller (2004) as a continuous counterpart of the discrete-time generalized autoregressive conditional heteroskedastic (GARCH hereafter) model proposed by Bollerslev (1986). The idea is to develop in continuous time a model that is able to capture some stylized facts observed in financial time series exploiting only one source of randomness for returns and for variance dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…Starting from the observation that the variance of a GARCH(p, q) is an autoregressive moving average model (ARMA hereafter) with order (q, p − 1) (see Brockwell et al 2006, for more details), the variance is modeled with a continuous-time autoregressive moving Average process (CARMA hereafter) with order (q, p − 1) (see Brockwell 2001;Tómasson 2015;Brockwell, Davis, and Yang 2007, and many others) driven by the discrete part of the quadratic variation of the Lévy process in the returns. Although this representation is different from that used by Klüppelberg et al (2004) for the COGARCH(1, 1) process, the latter can be again retrieved as a special case.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

COGARCH (p,q) : Simulation and Inference with the yuima Package

Iacus¹,

Mercuri²,

Rroji³

2017

J. Stat. Soft.

View full text Add to dashboard Cite

In this paper we show how to simulate and estimate a COGARCH(p, q) model in the R package yuima. Several routines for simulation and estimation are introduced. In particular, for the generation of a COGARCH(p, q) trajectory, the user can choose between two alternative schemes. The first is based on the Euler discretization of the stochastic differential equations that identify a COGARCH(p, q) model while the second considers the explicit solution of the equations defining the variance process.Estimation is based on the matching of the empirical with the theoretical autocorrelation function. Three different approaches are implemented: minimization of the mean squared error, minimization of the absolute mean error and the generalized method of moments where the weighting matrix is continuously updated. Numerical examples are given in order to explain methods and classes used in the yuima package.

show abstract

Section: Cogarch(p Q) Models Driven By a Lévy Processmentioning

confidence: 99%

“…As shown in Klüppelberg et al (2004) and remarked in Brockwell et al (2006), the condition in (6) is also necessary for the COGARCH(1, 1) case and can be simplified as:…”

Section: Cogarch(p Q) Models Driven By a Lévy Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

COGARCH (p,q) : Simulation and Inference with the yuima Package

Iacus¹,

Mercuri²,

Rroji³

2017

J. Stat. Soft.

View full text Add to dashboard Cite

show abstract

“…The two processes ~(~1 , B (~) are independent Brownian motions. A different approach has been taken by Kliippelberg, Lindner and Maller [27], who started with a discrete-time GARCH(1,l) model and replaced the noise variables by a LBvy process L with jumps ALt = Lt -Lt-, t > 0. This yields a stochastic volatility model of the type where p > 0 and V is left-continuous.…”

Section: Introductionmentioning

confidence: 99%

Extremal behavior of stochastic volatility models

Self Cite

View full text Add to dashboard Cite

Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility hassometimes quite substantialupwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by LBvy driven volatility processes as, for instance, by LBvy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving LBvy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,l) model. Driven by an arbitrary LBvy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels.

show abstract

GARCH Models

Francq¹,

Zakoïan²

2010

309

View full text Add to dashboard Cite

This paper considers the statistical inference of the class of asymmetric power-transformed GARCH(1,1) models in presence of possible explosiveness. We study the explosive behavior of volatility when the strict stationarity condition is not met. This allows us to establish the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the parameter, including the power but without the intercept, when strict stationarity does not hold. Two important issues can be tested in this framework: asymmetry and stationarity.The tests exploit the existence of a universal estimator of the asymptotic covariance matrix of the QMLE. By establishing the local asymptotic normality (LAN) property in this nonstationary framework, we can also study optimality issues.

show abstract

A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

Cited by 86 publications

References 16 publications

COGARCH (p,q) : Simulation and Inference with the yuima Package

COGARCH (p,q) : Simulation and Inference with the yuima Package

Extremal behavior of stochastic volatility models

GARCH Models

Contact Info

Product

Resources

About