Journal of the
Korean Mathematical Society JKMS

The classical Geometric Brownian motion (GBM) model for the price of a risky asset, from which the huge financial derivatives industry has developed, stipulates that the log returns are iid Gaussian. However, typical log returns data show a distribution with much higher peaks and heavier tails than the Gaussian as well as evidence of strong and persistent dependence. In this paper we describe a simple replacement for GBM, a fractal activity time Geometric Brownian motion (FATGBM) model based on fractal activity time which readily explains these observed features in the data. Consequences of the model are explained, and examples are given to illustrate how the self-similar scaling properties of the activity time check out in practice.

Keywords: risky asset model, geometric Brownian motion, heavy tails, long range, dependence, fractal activity time, self-similarity

MSC numbers: Primary 91B84, 91B28; Secondary 62M10, 60G18