A guaranteed deterministic approach to superhedging: no arbitrage market condition

For a discrete-time superreplication problem, a guaranteed deterministic formulation is considered: the problem is to ensure a complete coverage of the contingent claim on an option under all scenarios which are set using a priori defined compacts, depending on the price history: price increments at each moment of time must lie in the corresponding compacts. The market is considered with trading constraints and without transaction costs. The statement of the problem is game-theoretic in nature and leads directly to the Bellman–Isaacs equations. In this article, we study several notions that formalize the “no arbitrage” property of the market in the context of the deterministic approach and study their properties. A new concept of robustness (structural stability) of the “no arbitrage” properties of the market is introduced.