We investigate indifference pricing under perturbations of preferences in small and large markets. We establish stability results for small perturbations of preferences, where the latter can be stochastic. We obtain a sharp condition in terms of the associated concave and convex envelopes and provide counterexamples demonstrating that, in general, stability fails. Next, we investigate a class of models where the indifference price does not depend on the preferences or the initial wealth. Here, under the existence of an equivalent separating measure, in the settings of deterministic preferences, we show that the class of indifference price invariant models is the class of models where the dual domain is stochastically dominant of the second order. We also provide a counterexample showing that, in general, this result does not hold over stochastic preferences, where instead, we show that the indifference price invariant models are complete models (in both small and large markets). In the process, we establish a theorem of independent interest on the stability of the optimal investment problem under perturbations of preferences. Our results are new in both small and large markets, and thus, in particular, we introduce large stochastically dominant models, give examples of such models, and characterize them as the indifference price invariant ones over deterministic preferences.