Stability and Dual Valuation of Contingent Claims under Rockafellian Perturbations

Abstract

We study the stability of solutions to the discrete-time contingent-claim problem over a finite investment horizon when uncertainty is modeled by random variables with finite discrete support. Our main contribution is to use Rockafellian perturbations as a framework for this stability analysis: we construct perturbations of the underlying probability distribution, of the contingent claim, and of both jointly, and we establish epi-convergence of the corresponding approximating Rockafellians for the primal problem. The associated hypo-convergent approximations yield stable dual problems which, in turn, imply convergence of the dual variables, interpreted as shadow prices. This analysis reveals a connection between the duality gap and the value of perfect information and it provides conditions under which strong duality holds. We also construct examples in which epi-convergence fails due to critical scenarios with vanishing probabilities but unbounded impacts, illustrating the boundary between well-behaved and ill-conditioned contingent-claim problems.