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Pricing equity-linked pure endowments via the principle of equivalent utility. (English) Zbl 1103.91370

Summary: We consider a pure endowment contract whose life contingent payout is linked to the performance of a risky stock or index. Because of the additional mortality risk, the market is incomplete; thus, a fundamental assumption of the Black–Scholes theory is violated. We price this contract via the principle of equivalent utility and demonstrate that, under the assumption of exponential utility, the indifference price solves a nonlinear Black–Scholes equation; the nonlinear term reflects the mortality risk and exponential risk preferences in our model. We discuss qualitative and quantitative properties of the premium, including analytical upper and lower bounds.

MSC:

91B30 Risk theory, insurance (MSC2010)
49L20 Dynamic programming in optimal control and differential games
60H30 Applications of stochastic analysis (to PDEs, etc.)
91B28 Finance etc. (MSC2000)
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