Mathematical Model of Stock Prices via a Fractional Brownian Motion Model with Adaptive Parameters

The paper presents a mathematical model of stock prices using a fractional Brownian motion model with adaptive parameters (FBMAP). The accuracy index of the proposed model is compared with the Brownian motion model with adaptive parameters (BMAP). The parameters in both models are adapted at any time. The ADVANC Info Service Public Company Limited (ADVANC) and Land and Houses Public Company Limited (LH) closed prices are concerned in the paper. The Brownian motion model with adaptive parameters (BMAP) and fractional Brownian motion model with adaptive parameters (FBMAP) are applied to identify ADVANC and LH closed prices. The simulation results show that the FBMAP is more suitable for forecasting the ADVANC and LH closed price than the BMAP. 1. Introduction The ideas of using a Brownian motion process to explain the behavior of the risky asset prices were presented by Black et al. [1–3]. The stock prices presented in the paper are also the type in the risky asset prices. Therefore, the Brownian motion is usually used to model a stock price. However, Brownian motion process has the independent increments property. This means that the present price must not affect the future price. In fact, the present stock price may influence the price at some time in the future. Hence, Brownian motion process is not suitable to explain the stock price. Another process, a fractional Brownian motion process, exhibits a long range dependent property. Therefore, a fractional Brownian motion process can be used to describe the behavior of stock price instead of Brownian motion process. The rate of return and volatility in general asset pricing model are usually the constant parameters. Actually, the rate of return and volatility in the model are not constant at any time. In the paper, these parameters are updated depending on time by using the new information. The ADVANC Info Service Public Company Limited (ADVANC) and Land and Houses Public Company Limited (LH) stock prices are considered in the paper. These two stocks are chosen from different stock exchange of Thailand (SET) industry groups. The ADVANC and LH prices are selected from technology group (TECH) and property and construction group (PROPCON), respectively. The ADVANC and LH stock price models are studied by using fractional Brownian motion process to explain uncertainly behavior instead of Brownian motion process. The Brownian motion model with adaptive parameters (BMAP) and the fractional Brownian motion model with adaptive parameters (FBMAP) are presented to model the ADVANC and LH stock prices. The paper is