Quasi-likelihood estimation in a mixed fractional Black-Scholes model

Let BH=ₜH, t ≥ 0\ be a fractional Brownian motion with Hurst index $0<H<1$, and let B=ₜ, t ≥ 0\ be an independent Brownian motion. In this study, we investigate the parameter estimation of a mixed fractional Black-Scholes model SₜH=S₀H+μ ∫₀ᵗSₛHds+σ ∫₀ᵗSₛHd(Bₛ+BₛH),where σ>0, μ∈ R are two unknown parameters. Using quasi-likelihood estimation, when the system is observed at some discrete time instants ᵢ=ih,i=0,1,2,…,n\, we give estimations of the parameters μ and σ provided h=h(n)→ 0, nh→ ∞ and h1+γn→ 1 for some γ>0, as n→ ∞. We present the asymptotic normality of the estimators based on the velocity of nh1+γ-1 tending to zero as n tends to infinity. Finally, we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.