Quasi-Likelihood Estimation in the Fractional Black–Scholes Model

In this paper, we consider the parameter estimation for the fractional Black–Scholes model of the form StH=S0H+μ∫0tSsHds+σ∫0tSsHdBsH, where σ>0 and μ∈R are the parameters to be estimated. Here, BH={BtH,t≥0} denotes a fractional Brownian motion with Hurst index 0<H<1. Using the quasi-likelihood method, we estimate the parameters μ and σ based on observations taken at discrete time points {ti=ih,i=0,1,2,…,n}. Under the conditions h=h(n)→0, nh→∞, and h1+γn→1 for some γ>0, as n→∞, the asymptotic properties of the quasi-likelihood estimators are established. The analysis further reveals how the convergence rate of nh1+γ−1 approaching zero affects the accuracy of estimation. To validate the effectiveness of our method, we conduct numerical simulations using real-world stock market data, demonstrating the practical applicability of the proposed estimation framework.