Risk-Minimization for Life Insurance Liabilities

In this paper we study the pricing and hedging of a very general class of life insurance liabilities by means of the risk-minimization approach. We find the price and risk-minimizing strategy in two cases, first in the case when the financial market consists only of one risky asset, e. g., a stock, and a bank account, and second in an extended financial market, allowing for investments in two additional traded assets, representing the systematic and unsystematic mortality risk. We also provide an application in the case of a unit-linked term insurance contract in a jump-diffusion model for the stock price and affine stochastic mortality intensity. The main novelties of this work are that we allow for hedging of the risk inherent in the insurance liabilities by investing not only in the stock and money market account but also in a longevity bond, representing the systematic mortality risk, and a pure endowment contract, accounting for the unsystematic mortality risk. Besides that we consider a very general setting regarding the underlying asset price and the structure of the insurance payment process studied; i.e., we work outside the Brownian setting, and in particular the asset price may have jumps. Finally, we are able to relax certain technical assumptions, such as the existence of the mortality intensity, and we do not require the independence of the underlying processes.