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- Title
A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate.
- Authors
Lu, Yaowen; Dang, Duy‐Minh
- 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 ε$$ \varepsilon $$‐monotone Fourier pricing method, where ε>0$$ \varepsilon >0 $$ 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 ε→0$$ \varepsilon \to 0 $$. 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.
- Subjects
INTEREST rates; SEPARATION of variables; GREEN'S functions; VISCOSITY solutions; HAMILTON-Jacobi-Bellman equation; PRICES; INTEGRO-differential equations
- Publication
Numerical Methods for Partial Differential Equations, 2024, Vol 40, Issue 3, p1
- ISSN
0749-159X
- Publication type
Article
- DOI
10.1002/num.23075