Modeling Behavior of Stock Price Using Stochastic Differential Equation with Stochastic Volatility

The purpose of this article is modeling the behavior of stock price using stochastic differential equations. The data for this study include daily observations of the total stock market index, the index of the top 50 companies and the index of the 30 largest companies in the Tehran Stock Exchange. The data are daily from March 25, 2006 to April 15, 2015.The geometric Brownian motion and geometric Brownian motion with nonlinear GARCH are used to modeling the behavior of price index. The results of this study includes the following: (1) According to the log likelihood function, geometric Brownian motion with nonlinear GARCH in the three groups studied data has better performance than the geometric Brownian motion. (2) Based on the model of stochastic differential equations with stochastic volatility, the total market index is more influenced by the good news. (3) The impact of the bad news on the index of the 30 largest companies is more than the impact of the good news. (4) The unconditional variance of the total stock market index has two structural breaks; the unconditional variance of index of the top 50 companies has one structural break and no structural breaks in the unconditional variance of index of the 30 largest companies.