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

TL;DR

This work develops a comprehensive hybrid estimation framework for a mixed fractional Black-Scholes model with random effects using discrete-time observations. It combines generalized method of moments for global parameters with per-subject plug-in estimation of random effects, and introduces a nonparametric distribution estimator based on Lagrange interpolation at Chebyshev-Gauss nodes. The authors prove strong consistency and joint asymptotic normality for the fixed parameters under a two-step asymptotic regime, and derive various asymptotic results for the random effects estimators and their distribution, including a fast-converging, boundary-stable estimator for the random-effects distribution. The approach is validated through numerical simulations and an empirical crypto-market application, demonstrating accurate recovery of memory, volatility, and heterogeneous drift components across assets.

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.

Figures (10)

• Figure 1: Qualitative comparison between the proposed estimator $\hat{F}_{m,n,N}$ and the Kernel estimator $\hat{F}_h$ for the distribution $\mathcal{B}(2,2)$.
• Figure 2: Qualitative comparison between the proposed estimator $\hat{F}_{m,n,N}$ and the Kernel estimator $\hat{F}_h$ for the distribution $\mathcal{G}(2,1)$.
• Figure 3: Qualitative comparison between the proposed estimator $\hat{F}_{m,n,N}$ and the Kernel estimator $\hat{F}_h$ for the distribution $\mathcal{N}(0.5,0.25)$.
• Figure 4: Qualitative comparison between the proposed estimator $\hat{F}_{m,n,N}$ and the Kernel estimator $\hat{F}_h$ for the distribution $0.5*\mathcal{N}(-2,1) + 0.5* \mathcal{N}(3,0.5)$.
• Figure 5: Qualitative comparison between the mean ISE of both estimators for each distribution function