An Empirical Characteristic Function Approach to VaR Under a Mixture-of-Normal Distribution with Time-Varying Volatility

Abstract

Calculation of risk measures, such as Value-at-Risk and expected shortfall, requires knowledge of the underlying asset’s or portfolio’s returns distribution. To be realistic, this must be allowed to change over time. GARCH can be a good way to model the random evolution of an asset’s volatility, but standard GARCH assumes the innovation at each time step comes from a normal distribution. The resulting conditionally Gaussian returns therefore have normal, not fat, tails. One way to fatten the tails of the returns distribution is to use a fat-tailed forcing process, such as a Student-t with a low number of degrees of freedom. An alternative approach is to model the returns process as a mixture of normals, but if the distributions that are mixed have constant parameters, the time variation in volatility disappears. In this article, Xu and Wirjanto describe how timevarying fat-tailed densities can be formed by mixing GARCH processes together. Performance comparisons against other models in calculating the tail risks for exchange rates on four currencies show that the GARCH mixture model works very well. <b>TOPICS:</b>Derivatives, tail risks, VAR and use of alternative risk measures of trading risk