Modelling Scale-Consistent VaR with the Truncated Lévy Flight

Returns in financial assets show consistent excess kurtosis, indicating the presence of large fluctuations not predicted by Gaussian models. Mandelbrot (1963) first proposed the idea that price changes distributed according to a Lévy stable law. The unique feature of Lévy-stable distributions in general is that they are stable under addition. However, these distributions have power law tails that decay too slowly from the point of view of financial modelling. In recent studies the truncated Lévy Flight has been shown to eliminate this problem and to be very promising for the modelling of financial dynamics. An exponential decay in the tails ensures that all moments are finite and the distribution is fat-tailed for short time scales and converges in a Gaussian process for increasing time scales, a feature observed in financial data. We propose a model with time varying scale parameter (GARCH process) with error terms that are truncated Lévy distributed. We determine the appropriate GARCH specification for each data set by conducting a specification test based on a generalization of the augmented GARCH process of Duan (1997). The Lévy flight includes a method for scaling up a single-day volatility to a multi-day volatility, precisely a ?-root-of-time rule, where ? is the characteristic parameter of the process. We use this rule to forecast future volatility and as a result estimate Value-at-Risk (VaR) several days ahead and compare it to the RiskMetricsTM (1996) approach, which is a special case of our method. We compare the models in in-sample- and out-of-sample analyses for a sample of stock index returns.