Abstract
This paper deals with the problem of estimating the unknown parameters in a long-memory process based on the maximum likelihood method. The mean-square and the almost sure convergence of these estimators based on discrete-time observations are provided. Using Malliavin calculus, we present the asymptotic normality of these estimators. Simulation studies confirm the theoretical findings and show that the maximum likelihood technique can effectively reduce the mean-square error of our estimators.
Highlights
Brownian motion has been widely used in the Black–Scholes option-pricing framework to model the return of assets
A crucial problem with the applications of these option-pricing formulas in the fractional Black– Scholes market in practice is how to obtain the unknown parameters in geometric fractional Brownian motion
The gfBm has been shown by many authors to be a useful description of financial time series
Summary
Brownian motion has been widely used in the Black–Scholes option-pricing framework to model the return of assets. A crucial problem with the applications of these option-pricing formulas in the fractional Black– Scholes market in practice is how to obtain the unknown parameters in geometric fractional Brownian motion (hereafter gfBm). Otherwise, applying these models with long memory and self-similarity requires efficient and accurate synthesis of discrete gfBm. Even though fBm has stationary self-similar increments, it is neither independent nor Markovian. In this paper, inspired by Hu et al [39] and Bertin et al [32], we estimate the unknown parameters of a special long-memory process with discrete-time data, namely gfBm, based on MLE. The fBm {BHt , t ∈ R} with the Hurst parameter H ∈ (0, 1) is a zero mean Gaussian process with covariance
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