Sensitivity analysis of utility-based prices and risk-tolerance wealth processes

In the general framework of a semimartingale financial model and a utility function U defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a “small” number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases: 1. for any utility function U, if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance; 2. for any financial model, if and only if U is a power utility function (U is an exponential utility function if it is defined on the whole real line).