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Probability boxes, also known as $p$-boxes, correspond to sets of probabilitydistributions bounded by a pair of distribution functions. They fall into theclass of models known as imprecise probabilities. One of the central questionsrelated to imprecise probabilities are the intervals of values corresponding toexpectations of random variables, and especially the interval bounds. Ingeneral, those are attained in extremal points of credal sets, which denoteconvex sets of compatible probabilistic models. The aim of this paper is acharacterization and identification of extreme points corresponding to$p$-boxes on finite domains. To accomplish this, we utilize the concept ofnormal cones. In the settings of imprecise probabilities, those correspond tosets of random variables whose extremal expectations are attained in a commonextreme point. Our main results include a characterization all possible normalcones of $p$-boxes, their relation with extreme points, and the identificationof adjacency structure on the collection of normal cones, closely related tothe adjacency structure in the set of extreme points.

A complete characterization of normal cones and extreme points for $p$-boxes