In this article, insurance claims are priced using an indifference pricing principle. We first revisit the traditional economic framework and then extend it to incorporate a financial (sub)market as a tool to invest and to (partially) hedge. In this context, we derive lower bounds for claims’ prices, and these bounds correspond to the market prices of some explicitly known financial payoffs. In particular, we show that the discounted expected value is no longer valid as a classical lower bound for insurance prices in general: it has to be corrected by a covariance term that reflects the interaction between the insurance claim and the financial market. Examples that deal with equity‐linked insurance contracts illustrate the article.