Modeling fat tails in stock returns: a multivariate stable-GARCH approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. Returns on assets are assumed to follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student’s t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions of our model to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.