The distributional form of financial asset returns has important implications for the theoretical and empirical analyses in economics and finance. It is now a well-established fact that financial return distributions are empirically nonstationary, both in the weak and the strong sense. One first step to model such nonstationarity is to assume that these return distributions retain their shape, but not their localization or size (volatility ) as the classical Gaussian distributions do. In that case, one needs also to pay attention to skewedness and kurtosis, in addition to localization and size. This modeling requires special Zolotarev parametrizations of financial distributions, with four parameters, one for each relevant distributional moment. Recently popular stable financial distributions are the Paretian scaling distributions, which scale both in time T and frequency . For example, the volatility of the lognormal financial price distribution, derived from the geometric Brownian asset return motion and used to model Black-Scholes (1973) option pricing, scales according to T0.5. More generally, the volatility of the price return distributions of Calvet and Fishers (2002) Multifractal Model for Asset Returns (MMAR) scales according to , where the Zolotarev stability exponent z measures the degree of the scaling, and thus of the nonstationarity of the financial returns.

As we discussed in Los (2005a), the distributional form of financial asset returns has important implications for the theoretical and empirical analyses in economics and finance. For example, asset, portfolio and option pricing theories are typically based on the shape of these distributions, which some researchers have tried to recover from financial market prices. For example, Jackwerth and Rubinstein (1996) and Melick and Thomas (1997) did this for the options markets.

Stable distributions retain their shape over time, but not necessarily their size. They, long after having been in fashion for a short-lived period in the 1960s are currently en vogue for risk valuation, asset and option pricing, and portfolio management. They provide much more realistic financial risk profiles, not only in the high frequency FX markets, where, for example, excess kurtosis is found, but also in the persistent stock markets (Hsu, Miller and Wichern, 1974; Mittnik and Rachev, 1993a and b; Chobanov et al., 1996; McCulloch, 1996; Cont, Potters and Bouchaud, 1997; Gopikrishnan et al., 1998; Muller et al., 1998; Los, 2000).