A unified pricing of variable annuity guarantees under the optimal stochastic control framework

TL;DR

This paper introduces a unified stochastic control framework for pricing variable annuity guarantees, developing an efficient numerical method and providing benchmark results for practitioners.

Contribution

It presents a novel direct integration numerical method using Gauss-Hermite quadrature and splines, improving efficiency over PDE methods for pricing guarantees.

Findings

The new method is highly efficient compared to PDE approaches.

Numerical results for GMAB guarantees serve as benchmarks.

The framework unifies pricing of various variable annuity guarantees.

Abstract

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.