Fair pricing and reserving of variable annuities with guarantees under the benchmark approach

In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimum withdrawal benefits. Among three models of the equity index, we consider two pricing approaches, the classical risk-neutral approach and the real-world pricing under 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 real-world pricing is achieved using the benchmark approach, introduced by Platen (2001. A minimal financial market model. In Trends in mathematics (pp. 293–301). Birkhäuser). 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 and third models are the Black–Scholes model and the Heston model for the equity index, where the pricing of contracts is performed within the risk-neutral framework. We demonstrate that when the insurer prices and reserves using the minimal market model, the reserves are significantly lower than those under either the Black–Scholes or the Heston model. Furthermore, when the insured employs a dynamic withdrawal strategy based on the minimal market model, the total withdrawal amount exceeds that received when using either the Black–Scholes or the Heston model. We employ a novel approximation to the valuation of the VA having GMWB features, which permits fast valuation and multiple backtests of reserving strategies to be performed that would otherwise be impractical.