Pricing and Hedging Variable Annuities in a Lévy Market: A Risk Management Perspective

Pricing and hedging life insurance contracts with minimum guarantees are major areas of concern for insurers and researchers. In this article, we propose a unified framework for pricing, hedging, and assessing the risk embedded in the guarantees offered by Variable Annuities in a Lévy market. We address these questions from a risk management perspective. This method proves to be fast, accurate, and efficient. For hedging, we use a local risk minimization to provide a concise formula for the optimal hedging ratio. We also consider hedging strategies that use a portfolio of standard options. For assessing risk, we introduce an accumulated discounted loss function that takes mortality, transaction costs, and fees into account. We apply our resulting unified framework to the Minimum Guarantees for Maturity Benefit, Death Benefit, and Accumulation Benefit contracts. We illustrate the whole method with CGMY and Kou processes, which prove to offer a realistic modeling for financial prices. From this application, we draw important practical implications. In particular, we show that the assumption of geometric Brownian motion leads to undervalue the actual economic capital necessary to hedge and gives an illusion of safety.