This paper develops a framework for valuing mortality-contingent claims, specifically life annuities, in an incomplete market where mortality risk is non-diversifiable. The authors assume that the insurer requires compensation for bearing this risk, quantified by a pre-specified instantaneous Sharpe ratio. The valuation proceeds by constructing a portfolio consisting of the annuity liability and a self-financing sub-portfolio of bonds and cash, then choosing the bond holding to minimize the local variance (local risk minimization). The annuity's value is then determined so that the portfolio's instantaneous Sharpe ratio equals a given constant α.

For a single life annuity, the value solves a non-linear partial differential equation (PDE). The authors show that this value coincides with the upper good deal bound of Cochrane and Saá-Requejo (2000), providing a new interpretation. As the number of contracts n increases, the value per contract converges to a limit that satisfies a linear PDE, which can be expressed as an expectation under an equivalent martingale measure. This limiting result demonstrates that the instantaneous Sharpe ratio acts as the market price of mortality risk. The paper also compares the approach with indifference pricing and discusses the case α = 0, which recovers the local risk minimization price.