In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimum withdrawal benefits. We consider two pricing approaches, the classical risk-neutral approach and the benchmark approach, and we examine the associated static and optimal behaviors of both the investor and insurer. The first model considered is the so-called minimal market model, where pricing is achieved using the benchmark approach. The benchmark approach was introduced by Platen in 2001 and has received wide acceptance in the finance community. Under this approach, valuing an asset involves determining the minimum-valued replicating portfolio, with reference to the growth optimal portfolio under the real-world probability measure, and it both subsumes classical risk-neutral pricing as a particular case and extends it to situations where risk-neutral pricing is impossible. The second model is the Black-Scholes model for the equity index, where the pricing of contracts is performed within the risk-neutral framework. Crucially, we demonstrate that when the insurer prices and reserves using the Black-Scholes model, while the insured employs a dynamic withdrawal strategy based on the minimal market model, the insurer may be underestimating the value and associated reserves of the contract.
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