Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

Abstract

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.

Keywords:

• GMWB
• Willow tree scheme
• Dynamic withdrawal strategies
• Hedging strategies

JEL Code:

• G12
• C63

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† We adopt the 1994 Group Annuitant Mortality (GAM) Static Table and 1994 Mortality Improvement Projection Scale from the Society of Actuaries Group to estimate the mortality.

Additional information

Funding

The works of Bing Dong and Wei Xu were partially supported by the Natural Science Foundation of China (Project Number: 71771175, U1811462) and the Fundamental Research Funds for the Central Universities. The work of Yue Kuen Kwok was supported by the General Research Fund (Project Number: 16302416) of the Hong Kong Research Grants Council.