Abstract
The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities , however, the probability density found does not exhibit the cubic inverse law.
Highlights
Physics has long been a large source of ideas for economics
Since the prices of stocks of companies exhibit unpredictable fluctuations, they can be modeled by stochastic differential equations which becomes very important in the pricing of financial derivatives [1,2]
Since the celebre equation for price dynamics of the European market derived by Black and Scholes [3] until nowadays, the modeling of financial has been focused to simulate the behavior of market structure, trading mechanism and price dynamics [4]
Summary
Physics has long been a large source of ideas for economics. Every investor would like to be able to predict the price of a stock in the same way as physicists predict the trajectory or position in function of time of a particle. Since the prices of stocks of companies exhibit unpredictable fluctuations, they can be modeled by stochastic differential equations which becomes very important in the pricing of financial derivatives [1,2]. A very important model in modelling of financial market is the Ising model, H = ∑hiji Jij σi σj It constitutes in a very general class of stochastic dynamical model developed to describe interacting elements, particles, and agents in physics and biology. The study of volatility is crucial to reveal the underlined mechanism of markets dynamics, and it is useful for traders because it helps them in estimating risk and optimization of portfolio [20] Another quantity of interest in price dynamics is the return r (t).
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