In this paper we investigate price and Greeks computation of a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity (VA) when both stochastic volatility and stochastic interest rate are considered together in the Heston Hull-White model. Specifically, we employ an improved version of the Hybrid Tree-PDE (HPDE) approach introduced by Briani et al.. Such a numerical method turns out to be particularly suitable to handle the long maturity of GMWB products and to solve the dynamic control problem due to computing the optimal withdrawal strategy. Then, in order to speed up the computation, we employ Gaussian Process Regression (GPR). Starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole price function on a defined domain. The regression algorithm consists of algorithm training and evaluation. The first step is the most time demanding, but it needs to be performed only once, while the latter is very fast and it requires to be performed only when predicting the target function. The developed method, as well as for the calculation of prices and Greeks, can also be employed to compute the no-arbitrage fee, which is a common practice in the Variable Annuities sector. Numerical experiments show that the accuracy of the values estimated by GPR is high, while the computational cost is much lower than the one required by a direct calculation by HPDE. Finally, we stress out that the analysis is carried out for a GMWB annuity but it could be generalized to other insurance products.
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