Some empirical studies for the applications of fractional G-Brownian motion in finance

Since the fractional G-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and G-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-G-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence (1/2 < H < 1) or antipersistence (0 < H < 1/2), but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different σ² and σ̄², but Hurst parameters H < 1/2, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.