🤖 AI Summary
This study investigates the stability of solutions to discrete-time contingent claims problems under perturbations of both the underlying probability distribution and the claim structure, assuming a finite discrete support. Employing the Rockafellian perturbation framework together with epi-convergence and hypo-convergence from variational analysis, the work establishes—for the first time—a systematic connection between the convergence of primal and dual problems, and uncovers an intrinsic relationship between the duality gap and the value of perfect information. Key contributions include sufficient conditions for strong duality, a proof of solution stability under reasonable perturbations, and the construction of counterexamples that delineate the critical boundary where epi-convergence fails, thereby precisely distinguishing well-posed from ill-posed instances of the problem.
📝 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.