Comparing the Empirical and the Theoretical Probability Density of Return in which Variance Obeying Ornstein-Uhlenbeck Process at Indonesian Stock Exchange IDX

Abstract

The distribution of the probability density of a return index with stochastic volatility has been calculated. Here the stock index is assumed to follow geometric Brownian motion, while the variance is assumed to obey Ornstein-Uhlenbeck process as in Heston model. The distribution of the probability density of the return which is obtained by solving the Fokker-Planck equation of two dimensional index and the variance have been compared with the probability density taken from Indonesian Stock Exchange (IDX). In this study, we use Jakarta Islamic Index (JII), LQ45 and Jakarta Composite Index (JCI) data series from 2004 to 2012. We have shown that the theoretical probability density of return obtained from the calculation is in agreement with the empirical probability density. The theoretical probability density with stochastic volatility is closer to the empirical one than that of the Gaussian, particularly at the tail. The variance probability density at stationary state can be obtained by fitting the empirical probability density obtained from IDX data series with an integral expression obtained from quantum mechanical method.