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

TL;DR

This paper introduces a novel semi-Lagrangian Fourier method for pricing GMWBs with continuous withdrawals under jump-diffusion and stochastic interest rates, ensuring convergence and capturing complex market features.

Contribution

It develops an $psilon$-monotone Fourier pricing scheme for high-dimensional HJB-QVI problems with jumps and stochastic rates, with proven convergence and practical effectiveness.

Findings

Impact of jumps and stochastic rates on GMWB prices analyzed

New numerical method demonstrates convergence to viscosity solution

Enhanced understanding of optimal withdrawal strategies under complex models

Abstract

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 $\epsilon \to 0$. We present a comprehensive study of the impact of simultaneously considering jumps in the sub-account 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.