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A Guaranteed Deterministic Approach to Superhedging: A Game Equilibrium in the Case of No Trading Constraints
A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure coverage of the possible payout under an option contract for all admissible scenarios. These scenarios are given by means of compacts given a priori, which depend on the prehistory of prices: the increments of the (discounted) price at each moment of time must lie in the corresponding compacts. The reference probability, common for financial mathematics, is not needed. In the current framework we arrive at a control problem under uncertainty with discrete time, which has a game-theoretic interpretation. The capital, which will be necessary at some moment in time to cover the contingent liability during the time interval up to expiration, satisfies Bellman–Isaacs equations for both the pure and the mixed strategies. The stochastic description of price dynamic therefore arises when considering mixed strategies of the “market,” such that the conditional distributions of price increments given price history have supports contained in the corresponding compacts. In the present paper, under the assumption of no trading constraints, we prove that if there are no arbitrage opportunities, then the equilibrium holds for mixed strategies with the conditional distribution of price increments concentrated in a finite set, and we show that the optimal strategies of the “market” can be found among the “risk-neutral” strategies.