Modeling Financial Markets Using Combined Ornstein-uhlenbeck Process with Levy Noise

The main purpose of this paper is to investigate a developed stochastic algorithm for modeling financial markets using the Ornstein-uhlenbeck process combined with Levy noise. Using the closing prices of stock markets, it can be concluded that the stochastic model of the Ornstein-uhlenbeck process with time-dependent parameters significantly improves the performance of stock price forecasting.

At first, we study the stochastic differential equation that is composed of Ornstein-uhlenbeck independent processes. Since these processes are extracted by the gamma process, we call it the gamma Ornstein-uhlenbeck process, We used a stochastic differential equation under the combination of two independent processes and simulate the time series data.  The parameter estimation is done using the maximum likelihood estimator.

To illustrate the performance of the proposed model, we apply the desired stochastic differential equation for a set of financial time series from Tehran Oil Refining Company, Saipa Azin, and the Cement of Urmia stock exchanges. The simulated data mimics the original financial time series data. This is observed from the estimates of root mean square error criteria.

Numerical results show that the predicted volatility of these companies is close to the simulated ones. The advantage of this methodology is the fact that the estimates obtained are stable around the true value and also the low errors indicate that the estimation procedure is accurate, therefore producing a higher forecasting accuracy. Thus, the proposed estimation algorithm is suitable with large data sets and has good convergence properties.