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

2004/08/31 by Claudia Klüppelberg, Alexander Lindner, Ross Maller · 11 citations
Economics, Econometrics and Finance · #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #Complex Systems and Time Series Analysis

paper · doi:10.1239/jap/1091543413

Abstract

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

Citations

• The Variance Gamma (V.G.) Model for Share Market Returns

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• Volatility Analysis with Realized GARCH-Ito Models
• Stock Market Volatility and Return Analysis: A Systematic Literature Review. [europepmc]
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