Unit-linked life insurance in Lévy-process financial markets - modeling, hedging and statistics

Abstract

The main aim of this thesis is the development of locally risk-minimizing hedging strategies for unit-linked life insurance contracts whose unit is modeled in a general Lévy-process financial market. It therefore merges the quite advanced and in recent years developed theory of Lévy-process financial markets with the theory of unit-linked life insurances and provides moreover a quadratic hedging framework for insurance payment streams exposed to pure financial and pure insurance risk. In summary this thesis is part of the ongoing research on the interface between financial and actuarial mathematics and generalizes the earlier work of Moeller (1998,2001) in the sense that the complete Black-Scholes financial market is replaced by an incomplete financial market in case of a more general geometric Lévy-driven model. We first derive the locally risk-minimizing hedging strategy and the associated hedging risk for a portfolio of unit-linked pure endowment and a portfolio of unit-linked term insurance contracts assuming that they are sold against a single premium at issuing date and that their insurance benefits are deferred, while earning interest, and cashed out at the end of the considered time horizon. In a subsequent chapter we show that locally risk-minimizing hedging also extends to hedging of payment streams if the Föllmer-Schweizer measure is used. Finally we derive, in a general Lévy-process financial market, locally risk-minimizing hedging strategies for general unit-linked life insurance contracts admitting intermediate premiums and benefits. Additionally we apply the asymmetric double exponential jump diffusion stock model to our setting and investigate its behavior under a Föllmer-Schweizer change of measure. The last chapter provides a parameter estimation method for this jump-diffusion model. It is based on the empirical characteristic function.