We introduce a Deep-PCA Copula-fARCH framework for modeling and forecasting functional volatility dynamics in high-dimensional financial time series. The proposed model integrates deep nonlinear representation learning with copula-based dependence modeling within a functional ARCH (fARCH) setting. Specifically, a convolutional¿recurrent autoencoder extracts low-dimensional latent features from squared functional returns, while a copula structure captures nonlinear dependence among these latent components. The conditional variance function is recursively updated in the learned latent space and reconstructed through the decoder network, enabling flexible modeling of heteroskedasticity across functional domains. We compare four variants of the framework¿FPCA fARCH, FPCA Copula-fARCH, Deep-PCA fARCH, and Deep-PCA Copula-fARCH¿through extensive real-data analysis on KOSPI intraday returns. Results demonstrate that the Deep-PCA based approaches consistently and significantly outperform traditional FPCA methods. Quantitatively, the Deep-PCA models achieve the lowest absolute errors (RMSE) and, importantly, a substantially higher directional accuracy (Correlation) compared to the best FPCA-based model. These findings establish the Deep-PCA Copula-fARCH model as the most robust and accurate approach for high-frequency volatility configuration, highlighting the advantages of combining deep representation learning with copula-based dependence modeling for functional volatility structures.
