Optimal Stopping Rules in the Valuation of Guaranteed Lifetime Withdrawal Benefits with Embedded Options

The main objective of this study is to develop the optimal stopping rules in the valuation of Guaranteed Lifetime Withdrawal Benefits (GLWB) in the presence of embedded options. GLWB is known as one of the most popular insurance products, which provides a life contingent income stream whilst permitting the policyholders to obtain benefits from the financial returns of investment vehicles. In the first part of our study, we discuss some identities regarding the first-exit times of geometric Brownian motion with affine drift (GBMAD). To be more precise, in this study, we provide an explicit solution to the Laplace transform of the first exit time of GBMAD to a fixed level. Note that GBMAD indicates the underlying stochastic process of the policyholder’s account value at a particular time for the GLWB contract. By employing Ito’s formula, we show that the Laplace transform corresponds to a second-order differential equation that is equivalent to Kummer’s equation. As a result, the solution is thus given explicitly in terms of gamma and confluent hypergeometric functions. These preliminary results will then be used to develop an analytical solution for the valuation of GLWB with embedded options. Next, in the second part of this study, we extend existing works on the valuation of GLWB contract by introducing a top-up option. This new rider provides a policyholder with an option to change the existing contract to a new one with an increased withdrawal rate and a decreased fee rate by paying an exercising cost proportional to the current account value. This option is the American type, which can be exercised at any time before the contract’s maturity date. In particular, we present an explicit solution to the optimal valuation for GLWB with an embedded top-up option from the policyholder’s and insurer’s perspectives. Firstly, from the policyholder’s viewpoint, the valuation is formulated as an optimal stopping problem aimed at determining an exercise time of the option and an optimal account level that maximizes the contract’s monetary value. By using our preliminary results in the first part of this thesis, we show that the optimal value function is given in terms of the confluent hypergeometric function, satisfying continuous and smooth pasting conditions. Following this, we establish the majorant and (super) harmonic properties of the value function to show the optimality of the solution. Secondly, the valuation from the insurer’s standpoint is also examined by employing an equivalence principle between the insurer’s liabilities and fee incomes to determine the fair value of the new fee rate. In the last part, we propose the optimal valuation of GLWB with a surrender option. Under this new feature, the policyholder has the right to terminate the contract, subject to paying a surrender fee proportional to the current account value. From the perspective of the policyholder, we formulate the valuation in terms of an optimal stopping problem of finding an optimal surrender time and an optimal account level at which the monetary value of the contract is maximized. Optimality of the solution is established using a martingale approach. The early exercise nature of the option implies that the fair value of the GLWB contract is indeed undervalued compared to the GLWB with a surrender option. Subsequently, we also discuss the valuation from the insurer’s perspective to find the required minimum initial endowment of the account value by applying the equivalence principle between the insurer’s total liabilities and fee incomes.