Parameter estimation for discretized geometric fractional Brownian motions with applications in Chinese financial markets

Abstract

It is widely accepted that financial data exhibit a long-memory property or a long-range dependence. In a continuous-time situation, the geometric fractional Brownian motion is an important model to characterize the long-memory property in finance. This paper thus considers the problem to estimate all unknown parameters in geometric fractional Brownian processes based on discrete observations. The estimation procedure is built upon the marriage between the bipower variation and the least-squares estimation. However, unlike the commonly used approximation of the likelihood and transition density methods, we do not require a small sampling interval. The strong consistency of these proposed estimators can be established as the sample size increases to infinity in a chosen sampling interval. A simulation study is also conducted to assess the performance of the derived method by comparing with two existing approaches proposed by Misiran et al. (International Conference on Optimization and Control 2010, pp. 573–586, 2010) and Xiao et al. (J. Stat. Comput. Simul. 85(2):269–283, 2015), respectively. Finally, we apply the proposed estimation approach in the analysis of Chinese financial markets to show the potential applications in realistic contexts.