  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.
