Stable Distributions

Many of the concepts in theoretical and empirical finance developed over thepast decades - including the classical portfolio theory, the Black-Scholes-Mertonoption pricing model and the RiskMetrics variance-covariance approach toValue at Risk (VaR) - rest upon the assumption that asset returns followa normal distribution. However, it has been long known that asset returnsare not normally distributed. Rather, the empirical observations exhibit fattails. This heavy tailed or leptokurtic character of the distribution of pricechanges has been repeatedly observed in various markets and may be quan-titatively measured by the kurtosis in excess of 3, a value obtained for thenormal distribution (Bouchaud and Potters, 2000; Carr et al., 2002; Guillaumeet al., 1997; Mantegna and Stanley, 1995; Rachev, 2003; Weron, 2004).