  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. We consider a numerical method the solves the dynamic control
problem due to the computing of the optimal withdrawal. Moreover, 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 with very low computational cost. Finally, we stress out that the analysis
is carried out for a GMWB annuity but it could be generalized to other
insurance products.
