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 S t H = S 0 H + μ ∫ 0 t S s H d s + σ ∫ 0 t S s H d B s H , where σ > 0 and μ ∈ R are the parameters to be estimated. Here, B H = { B t H , 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 { t i = i h , i = 0 , 1 , 2 , … , n } . Under the conditions h = h ( n ) → 0 , n h → ∞ , and h 1 + γ n → 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 n h 1 + 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.