Simulating Fractal Financial Markets

In general, the absolute majority of financial market models is based on the stochastic properties of the asset returns, while the properties of the related asset quantities play a minor role. Starting from these remarks, in this paper we propose a system of nonlinear and stochastic difference equations in which the asset price behaviour and the corresponding asset quantity one are jointly taken into account. More precisely, in order effectively to represent the properties of the real asset price variations, we assume that (also on the basis of well known empirical evidences) their dynamics is distinguished by different stochastic processes alternating each other: the “classical” standard Brownian one, the fractional Brownian motion (which is able to represent the dependence among the returns), and the Pareto-Lévy stable one (which is able to represent the the non-Gaussian distributional features). All these processes are characterized by the same “fractal” quantity, the exponent of Hurst, which is properly utilized in the proposed dynamical model in order to represent the different stochastic properties of the asset price changes. Finally, because of the possible “bad” analytical peculiarities of the system itself, we investigate its dynamics by means of an agent-based approach developed in the Swarm software environment.