Two-dimensional stochastic dynamics as model for time evolution of the financial market

Abstract

• The long tail distribution of the probability density of the retorns is verified and the power law of the long tail distribution was obtained. • The Hurst index is calculated using the Detrended fluctuation analysis (DFA) analysis. • The Black & Scholes equations for the European call are determined. The dynamical model of coupled prices in the financial market is proposed based in a set of coupled stochastic differential equations where we assume there are two stocks, each with stochastic differential equation. These stocks are typically correlated. We analyse stylized facts observed in the financial markets as the long-tail distribution of the probability density of the returns and verified if the long tail of probability density distribution obeys to a scale law obeyed by the financial markets and therefore, if the model is adequate for the financial market. Furthermore, we obtain the behavior of the long range memory of the time series of the model and derive the correspondent Black-Scholes equation for the value of a European call.