  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.
