An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit

In this paper we explore an identity in distribution of hitting times of a finite variation process (integrated geometric Brownian motion) and a diffusion process (geometric Brownian motion with affine drift), both of which arise from various applications in financial mathematics. We develop semi-analytical solutions to fair charges of variable annuity guaranteed minimum withdrawal benefit from both a policyholder’s perspective and an insurer’s perspective. The pricing framework from the policyholder’s perspective was known previously in the literature only by numerical methods, whereas the insurer’s pricing method was used in the industry but only with Monte Carlo simulations. While comparing their similarities and differences, we prove under the assumption of no friction cost the two pricing approaches are equivalent. In the presence of friction cost, the semi-analytic solutions in this paper lead to a fast and accurate algorithm for determining rider charges and other management fees.