No Arbitrage in Insurance and the QP-Rule

This paper is an attempt to study fundamentally the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it via interest rates, or – as in hybrid products, equity-linked life insurance and variable annuities – directly to stocks or indices. By defining portfolio strategies on an insurance portfolio and combining them with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). A fundamental theorem provides two sufficient conditions for presence or absence of IFA, respectively. For the first one it utilizes the conditional law of large numbers and risk-neutral valuation. As a key result we obtain a simple valuation rule, called QP-rule, which is market consistent and excludes IFA. Utilizing the theory of enlargements of filtrations we construct a tractable framework for general valuation results, working under weak assumptions. The generality of the approach allows to incorporate many important aspects, like mortality risk or dependence of mortality and stock markets which is of utmost importance in the recent corona crisis. For practical applications, we provide an affine formulation which leads to explicit valuation formulas for a large class of hybrid products.