Pricing and Hedging GMWB Riders in a Binomial Framework

Abstract

The guaranteed minimum withdrawal benefit (GMWB) rider guarantees the return of premiums in the form of periodic withdrawals while allowing policyholders to participate fully in any market gains. The product has evolved into a lifetime version (GLWB) and is a vital component of the variable annuity marketplace, representing asset values of $294B as of September 2011.GMWB riders represent an embedded option on the account value with a fee structure that is different from typical financial derivatives. We present an in-depth study into pricing and hedging the GMWB rider from a financial economic perspective. Our main contributions are twofold. We construct a binomial asset pricing model for GMWBs under optimal policyholder behaviour which results in explicitly formulated perfect hedging strategies in a binomial world. The numerical toolbox for pricing GMWBs in a Black-Scholes world is expanded to include binomial methods.To motivate our work, we begin with a review of the continuous model and a comprehensive synthesis of results from the literature. Throughout, particular focus is placed on the unique perspectives of the insurer and policyholder and the unifying relationship. We also present an approximation algorithm 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 model. Finally, we explore the effect of deterministic mortality on pricing GMWBs, and run mortality simulations to obtain hedging results which support the diversification principle.

Valuation perspectives and decompositions for variable annuities with GMWB riders

“…In our view, the surrender charges complete the market and make the guarantees hedgeable. In fact, in the case of a binomial model for the product without mortality we prove in (Hyndman and Wenger, 2013b) that the product can be perfectly hedged with fee income and surrender charges.…”

“…Rather than directly applying a numerical approach to the continuous time results presented in this paper, which are of independent interest, we focus on the fundamental theory here and place an emphasis on numerical implementation in our subsequent works. In (Hyndman and Wenger, 2013b) we construct a complete binomial framework for the contract recovering similar valuation perspectives and decompositions to those presented in this paper, and include hedging results for the GMWB rider as well as numerical implementation techniques. In particular, in (Hyndman and Wenger, 2013b), we obtain numerical results based on the binomial model which can be viewed as an approximation to the model presented in this paper and which are in accordance with numerical results obtained using Monte-Carlo methods by (Kolkiewicz and Liu, 2012) and (Liu, 2010).…”

“…In (Hyndman and Wenger, 2013b) we construct a complete binomial framework for the contract recovering similar valuation perspectives and decompositions to those presented in this paper, and include hedging results for the GMWB rider as well as numerical implementation techniques. In particular, in (Hyndman and Wenger, 2013b), we obtain numerical results based on the binomial model which can be viewed as an approximation to the model presented in this paper and which are in accordance with numerical results obtained using Monte-Carlo methods by (Kolkiewicz and Liu, 2012) and (Liu, 2010). Further, while we disregard mortality in this paper we extend the binomial modelling framework to include mortality and obtain numerical results demonstrating the limits of hedging and diversification of mortality risk in (Hyndman and Wenger, 2013a).…”