As the understanding of GMxB-related risks deepens, insurance companies are increasingly seeking efficient annuity risk management systems. This paper is the first to extend the Karhunen-Loève (KL) expansion method to the pricing and Greeks estimation of the GMxB variable annuities written on multiple sub-account funds, under the multivariate Ornstein-Uhlenbeck stochastic volatility model. Additionally, the simulation-based path-wise (PW) and likelihood ratio (LR) methods are generalized for efficient Greeks computation within the multi-asset annuity framework. Through asymptotic analysis, we address a theoretical gap in the original KL expansion sampling framework. Numerical experiments demonstrate that the proposed method achieves computational efficiency and robustness, providing a practical and reliable framework for the risk management of complex multi-asset variable annuities.