A guaranteed deterministic approach to superhedging: the relationship between the deterministic and probabilistic problem statements without trading constraints

We consider a guaranteed deterministic approach to the discrete-time superreplication problem in which it is required to cover a contingent liability on a written option in all feasible scenarios. These scenarios are described by a priori given compact sets depending on the price history: at each time instant, the price increments must lie in the corresponding compact sets. We assume no transaction costs. The problem statement is game-theoretic and leads to the Bellman–Isaacs equations in pure and mixed “market” strategies. In the case of no trading constraints, we study the relationship between the Bellman functions in the “deterministic” and “probabilistic” statements of the superhedging problem. As established under very general conditions, the “probabilistic” Bellman function does not exceed the “deterministic” counterpart. Sufficient conditions for their coincidence are found.