The statistical properties of the time evolution of the price play a key role in the modeling of financial markets. For example, the knowledge of the stochastic nature of the price of a financial asset is crucial for a rational pricing of a derivative product issued on it. The full characterization of a stochastic process requires the knowledge of the conditional probability densities of all orders. This is an incredible task that cannot be achieved in practice. The usual empirical approach used by physicists is performed in two steps. The first concerns the investigation of time correlation and power spectrum, while the second concerns the study of the asymptotic pdf.

The most common stochastic model of stock price dynamics assumes that ln Y(t) is a diffusive process, and the ln Y(t) increments are assumed to be Gaussian distributed. This model, known as geometric Brownian motion, provides a first approximation of the behavior observed in empirical data. However, systematic deviations from the model predictions are observed, the empirical distributions being more leptokurtic than Gaussian distributions (Fig. 8.1). A highly leptokurtic distribution is characterized by a narrower and larger maximum, and by fatter tails than in the Gaussian case. The degree of leptokurtosis is much larger for high-frequency data (Fig. 8.2).

Based on theoretical assumptions and empirical analyses, several alternative models to geometric Brownian motion have been proposed.