Financial Asset Price Volatility Models and Empirical Analysis Based on Stochastic Differential Equations

Abstract

To accurately characterize the price volatility characteristics of financial assets, this paper uses the closing prices of the CSI 300 Index for 1488 trading days from 2018 to 2023 as a sample. Two types of stochastic differential equation models, basic and extended, are constructed. The basic model is centered on geometric Brownian motion and consists solely of the drift coefficient μ and the volatility coefficient σ. The extended model introduces dynamic jumps and stochastic volatility, accounting for both continuous fluctuations and extreme jumps. Using maximum likelihood estimation (MLE) for parameter estimation, data simulations show that the base model has μ=0.062 (annualized return 6.2%, p<0.01) and σ=0.185 (annualized -+-volatility 18.5%, p<0.001). All parameters of the extended model are significant (p<0.05), with an annualized jump of 2.4 times, a 5% increase in the probability of a jump for every 1-point increase in the VIX, and a 1.7-month return to normal volatility. Fitting and forecasting simulations validate the advantages of the extended model: the fitted RMSE is reduced by 40% compared to the base model (0.035 to 0.021), and the out-of-sample forecast RMSE is reduced by 40.48% (0.042 to 0.025). The fitted error for the extreme volatility point in October 2022 is less than 2%. Research shows that an expanded model incorporating jumps and stochastic volatility better aligns with the fluctuation patterns of the A-share market and can provide support for risk management.