Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products

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 on GMWB 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 pricing properties of the GMWB and performs a full mathematical analysis of the optimal dynamic withdrawal policies under the competing factors of time value of fund, optionality value provided by the guarantee and penalty charge on excessive withdrawal. When a proportional penalty charge is applied on any withdrawal amount, we can reduce the pricing formulation to an optimal stopping problem with lower and upper obstacles. We then derive the integral equations for the determination of a pair of optimal withdrawal boundaries. When a proportional penalty charge is applied 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 an immediate withdrawal of a finite amount. Under certain limiting scenarios such as a high policy fund value, the time close to expiry, or a low value of guarantee account, we manage to obtain analytical approximate solution to the singular stochastic control model of dynamic withdrawals.