Optimal Valuation of Variable Annuity Guaranteed Lifetime Withdrawal Benefits with Embedded Top-up Option

This paper generalizes earlier works on the valuation of variable annuity guaranteed lifetime withdrawal benefits (VAGLWB) to an extent in which the policyholders are given an option to change the parameters of the contract, i.e., the withdrawal and premium rates, at any time during the duration of the contract. Under this framework, we propose an embedded top-up option which gives the buyer an opportunity to change the contract to a new one with a larger withdrawal rate and reduced premium payment rate subject to paying a switching cost. The valuation is formulated in terms of the optimal stopping problem of finding an exercise time of the option which maximizes the contract value. We provide optimal solution to the stopping problem in which the asset’s price evolves according to a geometric Brownian motion. Moreover, we show that the value function of optimal stopping problem is given explicitly in terms of confluent hypergeometric function which satisfies continuous and smooth pasting conditions. Majorant and (super) harmonic properties of the value function are established to show optimality of the solution. Numerical examples are discussed to exemplify the main results and the sensitivity analysis of several parameters in the contract.