Hybrid estimation for a mixed fractional Black-Scholes model with random effects from discrete time observations

We propose a hybrid estimation procedure to estimate global fixed parameters and subject-specific random effects in a mixed fractional Black-Scholes model based on discrete-time observations. Specifically, we consider $N$ independent stochastic processes, each driven by a linear combination of standard Brownian motion and an independent fractional Brownian motion, and governed by a drift term that depends on an unobserved random effect with unknown distribution. Based on $n$ discrete time statistics of process increments, we construct parametric estimators for the Brownian motion volatility, the scaling parameter for the fractional Brownian motion, and the Hurst parameter using a generalized method of moments. We establish their strong consistency under the two-step regime where the observation frequency $n$ and then the sample size $N$ tend to infinity, and prove their joint asymptotic normality when $H \in \big(\frac12, \frac34\big)$. Then, using a plug-in approach, we consistently estimate the random effects, and we study their asymptotic behavior under the same sequential asymptotic regime. Finally, we construct a nonparametric estimator for the distribution function of these random effects using a Lagrange interpolation at Chebyshev-Gauss nodes based method, and we analyze its asymptotic properties as both $n$ and $N$ increase. We illustrate the theoretical results through a numerical simulation framework. We further demonstrate the efficiency performance of the proposed estimators in an empirical application to crypto returns data, analyzing five major cryptocurrencies to uncover their distinct volatility structures and heterogeneous trend behaviors.