A semi-Lagrangian ε-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate

We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump-diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no-arbitrage GMWB pricing problem as a time-dependent Hamilton-Jacobi-Bellman (HJB) Quasi-Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi-Lagrangian method and the Green's function of an associated linear partial integro-differential equation, we develop an epsilon-monotone Fourier pricing method, where epsilon > 0 is a monotonicity tolerance. Together with a provable strong comparison result for the HJB-QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB-QVI as epsilon -> 0. We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no-arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.