The Degree of Stability of Price Diffusion

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 (mean ) 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 a 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 T^{0.5}. More generally, the volatility of the price return distributions of Calvet and Fisher's (2002) Multifractal Model for Asset Returns (MMAR) scales according to T^{(1/(_{Z}))}, where the Zolotarev stability exponent _{Z} measures the degree of the scaling, and thus of the nonstationarity of the financial returns. Keywords: Stable distributions, price diffusion, stability exponent, Zolotarev parametrization, fractional Brownian motion, financial markets.