Analysis of optimal dynamic withdrawal policies in withdrawal guarantees products

The guaranteed minimum withdrawal benefits (GMWB) are popular riders in variable annuities with withdrawal guarantees. With withdrawals spread over the life of the annuities contract, the benefit promises to return the entire initial annuitization amount irrespective of the market performance of the underlying fund portfolio. Treating the dynamic withdrawal rate as the control variable, the earlier works have considered the construction of a continuous singular stochastic control model and the numerical solution of the resulting pricing model. This paper presents a more detailed characterization of the behavior of the GMWB price function and performs a full mathematical analysis of the optimal dynamic withdrawal policies under the competing forces of time value of fund and penalty charge on excessive withdrawal. When proportional penalty charge is applied on any withdrawal amount, we can reduce the pricing formulation to an obstacle problem with lower and upper obstacles. We then derive the integral equations for the determination of a pair of optimal withdrawal boundaries. When proportional penalty charge is applied only on the amount that is above the contractual withdrawal rate, we manage to characterize the behavior of the optimal withdrawal boundaries that separate the domain of the pricing models into three regions: no withdrawal, continuous withdrawal at the contractual rate and immediate withdrawal of finite amount. Under certain limiting conditions, we manage to obtain analytical approximate solution to the singular stochastic control model of dynamic withdrawal.