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

ArXiv ID: 2310.00606 “View on arXiv”

Authors: Unknown

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 $ε$-monotone Fourier pricing method, where $ε> 0$ 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 $ε\to 0$. 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.

Keywords: Impulse Stochastic Control, Hamilton-Jacobi-Bellman (HJB), Quasi-Variational Inequality (QVI), Jump-Diffusion, Fourier Pricing, Insurance / Derivatives

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats
  • Why: The paper relies heavily on advanced mathematics, including impulse stochastic control, HJB-QVI with cross derivatives, viscosity solutions, and a novel semi-Lagrangian Fourier method, with rigorous convergence proofs. However, it presents a numerical framework and impact analysis rather than backtest-ready implementation details, datasets, or empirical performance metrics like Sharpe ratios.
  flowchart TD
    A["Research Goal<br>Price continuous withdrawal GMWBs<br>under jump-diffusion & stochastic rates"] --> B{"Modeling & Formulation"}
    B --> C["Impulse Stochastic Control"]
    C --> D["Time-Dependent HJB-QVI<br>3 spatial dims, cross-derivatives"]
    D --> E{"Numerical Solution<br>Semi-Lagrangian Fourier Method"}
    E --> F["Discretization &<br>ε-Monotone Scheme"]
    F --> G["Data & Inputs<br>Market/Model Parameters<br>payoff structures"]
    G --> H["Computational Process<br>Solve PDE via<br>Green's Function & FFT"]
    H --> I["Key Findings/Outcomes<br>1. Pricing/Fee formulas<br>2. Optimal Withdrawal Strategy<br>3. Impact of Jumps & Rates"]