No Arbitrage in Insurance and the QP-Rule

22 Pages Posted: 23 Jun 2020

See all articles by Philippe Artzner

Philippe Artzner

University of Strasbourg - Institut de Recherche Mathématique Avancée, UMR 7501

Karl-Theodor Eisele

University of Strasbourg

Thorsten Schmidt

University of Freiburg

Date Written: May 22, 2020


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.

Keywords: arbitrage, fundamental theorem of asset pricing, insurance valuation, variable annuities, hybrid products, equity-linked life insurance, enlargement of filtration, affine processes, longevity risk, corona crisis, QP-rule

Suggested Citation

Artzner, Philippe and Eisele, Karl-Theodor and Schmidt, Thorsten, No Arbitrage in Insurance and the QP-Rule (May 22, 2020). Available at SSRN: or

Philippe Artzner

University of Strasbourg - Institut de Recherche Mathématique Avancée, UMR 7501 ( email )

7, rue René Descartes
Strasbourg, 67084

Karl-Theodor Eisele

University of Strasbourg ( email )

61, avenue de la foret noire
Strasbourg, Alsace 3000

Thorsten Schmidt (Contact Author)

University of Freiburg ( email )

Freiburg, D-79085

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