Guaranteed deterministic approach to superhedging: properties of binary European option

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 the present paper, we study the solution of these equations for the pricing in the case of binary European option in the framework of a multiplicative market model with no trading constrains. Certain properties of the solution are established and an algorithm of numerical solution of Bellman equation is proposed. From mathematical prospect, the interest is due to the discontinuity of the payout function.