GMWB Riders in a Binomial Framework - Pricing, Hedging, and Diversification of Mortality Risk

TL;DR

This paper develops a binomial model for GMWB riders in variable annuities, providing explicit hedging strategies, incorporating mortality risk, and demonstrating effective risk diversification through numerical examples.

Contribution

It introduces a novel binomial framework for GMWB pricing and hedging, including mortality risk, with improved computational efficiency and practical numerical illustrations.

Findings

Explicit hedging strategies funded by fee income.

Effective diversification of mortality risk.

Enhanced model accuracy with an Asian option approximation.

Abstract

We construct a binomial model for a guaranteed minimum withdrawal benefit (GMWB) rider to a variable annuity (VA) under optimal policyholder behaviour. The binomial model results in explicitly formulated perfect hedging strategies funded using only periodic fee income. We consider the separate perspectives of the insurer and policyholder and introduce a unifying relationship. Decompositions of the VA and GMWB contract into term-certain payments and options representing the guarantee and early surrender features are extended to the binomial framework. We incorporate an approximation algorithm for Asian options that significantly improves efficiency of the binomial model while retaining accuracy. Several numerical examples are provided which illustrate both the accuracy and the tractability of the binomial model. We extend the binomial model to include policy holder mortality and death benefits. Pricing, hedging, and the decompositions of the contract are extended to incorporate mortality risk. We prove limiting results for the hedging strategies and demonstrate mortality risk diversification. Numerical examples are provided which illustrate the effectiveness of hedging and the diversification of mortality risk under capacity constraints with finite pools.