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

Abstract: 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 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 strong consistency and joint asymptotic normality of these estimators. Then, from one trajectory, we consistently estimate the random effects, using a plug-in approach, and we study their asymptotic behavior under different asymptotic regimes as $N$ and $n$ grow. 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 the number of subjects $N$ and the number of observations per-subject $n$ increase. A numerical simulation framework is also investigated to illustrate the theoretical results of the estimators behavior.