Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate

In this paper, we investigate value and Greeks computation of a guaranteed minimum withdrawal benefit (GMWB) variable annuity, when both stochastic volatility and stochastic interest rate are considered together in the Heston–Hull–White model. In addition, as an insurance product, a guaranteed minimum death benefit is embedded in the contract. We consider a numerical method that 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), a machine learning technique that allows one to compute very fast approximations of a function from training data. In particular, starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole value 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.