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

TL;DR

This paper establishes a mathematical identity linking hitting times of certain stochastic processes and applies it to derive analytical solutions for valuing guaranteed minimum withdrawal benefits in variable annuities, improving upon previous numerical methods.

Contribution

It introduces a novel identity in distribution for hitting times of specific processes and applies it to obtain analytical valuation formulas for GMWB, bridging policyholder and insurer perspectives.

Findings

Analytical solutions for GMWB fair charge from policyholder's view.

Analytical solutions for GMWB fair charge from insurer’s view using complex inversion.

Proof of equivalence between two different pricing approaches under certain assumptions.

Abstract

In this paper we explore an identity in distribution of hitting times of a finite variation process (Yor's process) and a diffusion process (geometric Brownian motion with affine drift), which arise from various applications in financial mathematics. As a result, we provide analytical solutions to the fair charge of variable annuity guaranteed minimum withdrawal benefit (GMWB) from a policyholder's point of view, which was only previously obtained in the literature by numerical methods. We also use complex inversion methods to derive analytical solutions to the fair charge of the GMWB from an insurer's point of view, which is used in the market practice, however, based on Monte Carlo simulations. Despite of their seemingly different formulations, we can prove under certain assumptions the two pricing approaches are equivalent.