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

Changhong Guo; Shaomei Fang; Yong He; Yong Zhang; Changhong Guo; Shaomei Fang; Yong He; Yong Zhang

doi:10.3934/QFE.2025001

Quantitative Finance and Economics

2025, Volume 9, Issue 1: 1-39. doi: 10.3934/QFE.2025001

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Research article

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

1.
School of Management, Guangdong University of Technology, Yinglong Road 161, Guangzhou, Guangdong 510520, P. R. China
2.
Department of Mathematics, South China Agricultural University, Wushan Street 483, Guangzhou, Guangdong 510640, P. R. China

Received: 01 October 2024 Revised: 12 December 2024 Accepted: 06 January 2025 Published: 09 January 2025
JEL Codes: G15, G10

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 $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{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.
- fractional $ G $-Brownian motion,
- H-$ G $-normal distribution,
- volatility smoothness,
- RFSV-fGBm,
- parameter estimates
Citation: Changhong Guo, Shaomei Fang, Yong He, Yong Zhang. Some empirical studies for the applications of fractional $ G $-Brownian motion in finance[J]. Quantitative Finance and Economics, 2025, 9(1): 1-39. doi: 10.3934/QFE.2025001

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Abstract

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 $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{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.

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