A Hybrid Gaussian Process Regression Framework for Stable Volatility-Covariance Estimation: Evidence from Global Equity Indices

Accurate forecasting of the Volatility-Covariance Matrix (VCV) is central to regulatory capitaladequacy processes such as the Internal Capital Adequacy Assessment Process (ICAAP) and theComprehensive Capital Analysis and Review (CCAR). Traditional econometric models,including GARCH-family and Exponentially Weighted Moving Average (EWMA) approaches,suffer from parametric rigidity, distributional assumptions, and numerical instability under stress,leading to systematic underestimation of tail risk. This paper proposes and validates a novelHybrid Gaussian Process Regression–Historical Simulation (GPR-HS) framework for estimatingValue-at-Risk (VaR) and Expected Shortfall (ES) across a diversified portfolio of seven majorglobal equity indices.The framework decouples the VCV estimation problem: individual asset volatilities are modelleddynamically using Univariate GPR with a Matern 5/2 kernel, while inter-asset correlations areestimated via stable historical covariance. A key methodological contribution is the AggressiveNoise Initialization (ANI) strategy, which sets the initial White Noise kernel variance equal tothe empirical variance of the training returns, ensuring Gram matrix positive-definiteness,regularization, and conservative, regulatory-compliant forecasts. Evaluated using an expandingwindow forward-chaining cross-validation scheme over June 2020–June 2025, the GPR-HSframework achieves regulatory compliance in the majority of test splits; including a 100% ESpass rate at the portfolio level, while outperforming the static Historical VaR benchmark in71.4% of univariate cases by Quadratic Loss and 100% of cases by violation count. Keywords: Gaussian Process Regression, Value-at-Risk, Expected Shortfall, VolatilityCovariance Matrix, ICAAP, CCAR, Hybrid Framework, Aggressive Noise Initialization,Backtesting, Matern Kernel Replication code: https://colab.research.google.com/drive/1nrlSqmG10DNerNmEqGIh3EB9CcLWIgH9