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Abstract. Recently, crude oil price becomes volatile and have been the popular issue to be discussed in every country. Oil price fluctuations have major impact on the overall economy and finally will lead to increase in the inflation rate. It is important to describe these oil price fluctuations mathematically. This study aims to describe the above phenomena using geometric Brownian motion. Two crude oil prices, namely WTI and Brent have been analyzed based on daily oil price data from year 2000 until year 2015. Through the analysis using model assessment and model determination, crude oil price after year 2000 follows geometric Brownian motion process. We conclude that oil price fluctuations follow a geometric Brownian motion process without considering unexpected incidents.

Abstract. This paper studies the Ornstein-Uhlenbeck model that incorporates long memory stochastic volatility which is known as fractional Ornstein-Uhlenbeck model. The determination of the existence of long range dependence of the index prices of FTSE Bursa Malaysia KLCI is measured by the Hurst exponent. The empirical distribution of unobserved volatility is estimated using the particle filtering method. The performance between fractional Ornstein -Uhlenbeck and standard Ornstein -Uhlenbeck process had been compared. The mean square errors of the fractional Ornstein-Uhlenbeck model indicated that the model describes index prices better than the standard Ornstein-Uhlenbeck process.

Long memory and volatility have been used to measure risks associated with persistence in financial data sets. However, the persistence in volatility cannot be easily captured because some mathematical models are not able to detect these properties. To overcome this shortfall, this study develops a procedure to construct long-memory stochastic volatility (LMSV) model by using fractional Ornstein-Uhlenbeck (fOU) process in financial time series to evaluate the degree of persistence property of the data. Procedures for constructing the LMSV model and the estimation methods were applied to the real daily index prices of FTSE Bursa Malaysia KLCI over a period of 20 years. The least square estimator (LSE) and quadratic generalised variations (QGV) methods were used to estimate the drift and diffusion coefficient of the volatility process respectively. The long memory parameter was estimated by the detrended fluctuation analysis (DFA) method. The findings show that the volatility of the index prices exhibited a long memory process but the returns of the index prices did not show strong persistence properties. The root mean square errors (RMSE) obtained from various methods indicated that the performances of the model and estimators in describing returns of the index prices were good.