A guaranteed deterministic approach to superhedging: financial market model,trading constraints and Bellman–Isaacs equations

A guaranteed deterministic problem setting of super-replication with discrete time is proposed, as an alternative to the traditional probabilistic approach based on the use of the reference measure. Within the proposed framework, the reference measure is not needed, and aim of hedging of contingent claim is to guarantee coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The presentation focuses on achieving clarity, without aiming the greatest possible generality; this is the reason for the nature of a number of assumptions. The absence of transaction costs is assumed, the market is considered both with trade restrictions, and without trade restrictions. The game-theoretical approach immediately allows us to write down the corresponding Bellman–Isaac equations using economic interpretation of the problem.