Various parametric models have been developed to predict large volatility matrices, based on the approximate factor model structure. They mainly focus on the dynamics of the factor volatility with some finite high-order moment assumptions. However, the empirical studies have shown that the idiosyncratic volatility also has a dynamic structure and it comprises a large proportion of the total volatility. Furthermore, we often observe that the financial market exhibits heavy tails. To account for these stylized features in financial returns, we introduce a novel It^{o} diffusion process for both factor and idiosyncratic volatilities whose eigenvalues follow the vector auto-regressive (VAR) model. We call it the factor and idiosyncratic VAR-It^{o} (FIVAR-It^o) model. To handle the heavy-tailedness and curse of dimensionality, we propose a robust parameter estimation method for a high-dimensional VAR model. We apply the robust estimator to predicting large volatility matrices and investigate its asymptotic properties. Simulation studies are conducted to validate the finite sample performance of the proposed estimation and prediction methods. Using high-frequency trading data, we apply the proposed method to large volatility matrix prediction and minimum variance portfolio allocation and showcase the new model and the proposed method.
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