# Fair Pricing of Variable Annuities with Guarantees under the Benchmark Approach

## TL;DR

This paper compares the classical risk-neutral and benchmark approaches to pricing variable annuities with guarantees, revealing potential underestimation of reserves when using the Black-Scholes model with dynamic strategies.

## Contribution

It introduces a comparative analysis of pricing methods for variable annuities under different models, highlighting the implications of strategy and model choice on valuation.

## Key findings

- Benchmark approach extends classical pricing to complex scenarios.
- Black-Scholes model may underestimate reserves with dynamic strategies.
- Dynamic withdrawal strategies impact valuation accuracy.

## Abstract

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.