Variable annuities are life insurance products which are particularly attractive in a low interest rate environment. They allow the policyholder to participate in the growth of the economy through the performance of a reference fund, guaranteeing at the same time a minimum payment (GMAB) based on a fixed rate. A hybrid approach driven by time-inhomogeneous Lévy processes [1] is used to model the joint behavior of the underlying reference fund and the interest rate market. In addition to the choice at maturity between a guaranteed sum and the performance of a fund, the design of the contract incorporates a payment in case of death of the policyholder and surrender provisions. Following [2], the rate at which contracts are cancelled is modelled by a surrender intensity which depends on the performance of the reference fund, as well as the guaranteed rate, the interest rate prevailing at the time point of cancellation, and the surrender penalty function. Two versions of the surrender intensity are studied for valuation of the contract.

[1] Eberlein E. and Rudmann M. (2018), “Hybrid Lévy models: Design and computational aspects.” Applied Mathematical Finance, DOI: 10.1080/1350486X.2018.1536523.

[2] Escobar M., Krayzler M., Ramsauer F., Saunders D. and Zagst R. (2016), “Incorporation of stochastic policyholder behavior in analytical pricing of GMABs and GMDBs.” Risks, vol. 4(4), pp. 1-36.