Optimal valuation of variable annuity guaranteed lifetime withdrawal benefits with embedded top-up option

This paper generalizes earlier works on the variable annuity guaranteed lifetime withdrawal benefits (VAGLWB) by introducing an embedded top-up option to the contract. This new feature/rider gives the policyholder an option to top-up the existing contract to a new one with larger withdrawal rate and reduced premium rate subject to paying a cost proportional to the current account value. The option is of American type which can be exercised at anytime prior to the maturity of the contract. In this work, we provide an analytical solution to the risk-neutral valuation for the VAGLWB with embedded top-up option from both policyholder's and insurer's perspective. From the perspective of policyholder, the valuation is formulated in terms of an optimal stopping problem of finding an exercise time of the option and the optimal account level at which the monetary value of the contract is maximized. The optimal solution to the stopping problem is derived under geometric Brownian motion dynamics of the equity price, the underlying investment vehicle of VAGLWB. The optimal value function (early exercise premium of the option) is given explicitly in terms of the confluent hypergeometric function satisfying both continuous and smooth pasting conditions. Furthermore, majorant and (super) harmonic properties of the value function are established to show the optimality of the solution. In the absence of top-up option, i.e., the new contract has equal withdrawal and premium rates with that of the existing contract, the results reduce to that of Feng and Jing (2017). Valuation from the insurer's perspective is discussed using equivalence principle between insurer's liabilities and fee incomes to find the fair value of the new premium rate. Finally, numerical examples are provided to exemplify the main results.