Multivariate tempered stable model with long-range dependence and time-varying volatility

High-frequency financial return time series data have stylized facts such as the long-range dependence, fat-tails, asymmetric dependence, and volatility clustering. In this paper, a multivariate model which describes those stylized facts is presented. To construct the model, a multivariate ARMA-GARCH model is considered along with fractional Lévy process. The fractional Lévy process in this paper is defined by the stochastic integral with a tempered stable driving process. Parameters of the new model are fit to high-frequency returns for five U.S stocks. Approximated form of portfolio value-at-risk and average value-at-risk are provided and portfolio optimization is discussed under the model.