  We consider a class of stochastic programs whose uncertain data has an
exponential number of possible outcomes, where scenarios are affinely
parametrized by the vertices of a tractable binary polytope. Under these
conditions, we propose a novel formulation that introduces a modest number of
additional variables and a class of inequalities that can be efficiently
separated. Moreover, when the underlying polytope is the unit hypercube, we
present an extended formulation of polynomial size that can be solved directly
with off--the--shelf optimization software. We assess the advantages and
limitations of our formulation through a computational study.
