THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

Authors


  • This work was initiated during a visit of the authors to the Oxford-Man Institute at the University of Oxford in December 2007. The authors are very grateful for the hospitality and support. They thank Kevin Sheppard for a discussion leading them to consider (sup)OU models in the context of factor modeling. Furthermore, Ole Barndorff-Nielsen appreciates support from CREATES funded by the Danish National Research Foundation and Robert Stelzer gratefully acknowledges the support of the Technische Universität München-Institute for Advanced Study, funded by the German Excellence Initiative.

Robert Stelzer, Institute of Mathematical Finance, Ulm University, Helmhlotzstrasse 18, D-89069 Ulm, Germany; e-mail: robert.stelzer@uni-ulm.de.

Abstract

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second-order structure of the volatility, the log- returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well as the simple Ornstein–Uhlenbeck type stochastic volatility model behave under linear transformations. In particular, the models are shown to be preserved under invertible linear transformations. Finally, we discuss how (sup)OU stochastic volatility models can be combined with a factor modeling approach.

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