Volatility analysis for the GARCH–Itô–Jumps model based on high-frequency and low-frequency financial data

This paper introduces a model that can accommodate both the continuous-time-diffusion and discrete-time mixed-GARCH–Jump models by embedding the discrete mixed-GARCH-Jump structure in the continuous volatility process. The key feature of the proposed model is that the corresponding conditional integrated volatility adopts the mixed-GARCH-Jump structure that accounts for the effect of jumps on future volatility. A Griddy–Gibbs sampler approach is proposed to estimate parameters, and volatility forecasting and value-at-risk forecasting based on the peaks-over-threshold method are developed. Simulations are carried out to check the finite sample performance of the proposed methodology, and empirical studies show that, in general, volatility is heavily influenced by the continuous innovations, rather than the extreme reactions. We find that both the simulation and empirical results in most cases support the proposed model.