Estimating dynamic geometric fractional brownian motion and its application to long-memory option pricing

Abstract

Geometric fractional Brownian motion (GFBM) is an extended dynamic model of the traditional geometric Brownian motion, and has been used in characterizing the long term memory dynamic behavior of financial time series and in pricing long-memory options. A crucial problem in its applications is how the unknown parameters in the model are to be estimated. In this paper, we study the problem of estimating the unknown parameters, which are the drift µ, volatility s and Hurst index H, involved in the GFBM, based on discrete-time observations. We propose a complete maximum likelihood estimation approach, which enables us not only to derive the estimators of µ and s 2, but also the estimate of the long memory parameter, H, simultaneously, for risky assets in the dynamic fractional Black-Scholes market governed by GFBM. Simulation outcomes illustrate that our methodology is statistically efficient and reliable. Empirical application to stock exchange index with European option pricing under GFBM is also demonstrated. ©Dynamic Publishers, Inc.