The paper studies pricing of insurance products focusing on the pricing of annuities under uncertainty. This pricing problem is crucial for financial decision making and was studied intensively, however, many open questions still remain. In particular, there is a so-called "annuity puzzle" related to certain inconsistency of existing financial theory with the empirical observations for the annuities market. The paper suggests a pricing method based on the risk minimization such that both producer and customer seek to minimize the mean square hedging error accepted as a measure of risk. This leads to two different versions of the pricing problem: the selection of the annuity price given the rate of regular payments, and the selection of the rate of payments given the annuity price. It appears that solutions of these two problems are different. This can contribute to explanation for the "annuity puzzle".