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Guaranteed Deterministic Approach to Superhedging: Lipschitz Properties of Solutions of the Bellman–Isaacs Equations

  • Sergey N. Smirnov
Chapter
Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)

Abstract

For the discrete-time superreplication problem, a guaranteed deterministic formulation is proposed: the problem is to ensure the cheapest coverage of the contingent claim on an American option under all admissible scenarios. These scenarios are set by a priori defined compacts depending on the price history; the price increment at each moment of time must lie in the corresponding compact. The market is considered without trading constraints and transaction costs. The problem statement is game-theoretic in nature and leads directly to the Bellman–Isaacs equations of a special form under the assumption of no trading constraints. In the present study, we estimate the modulus of continuity of uniformly continuous solutions, including the Lipschitz case.

Keywords

Guaranteed estimates Deterministic price dynamics Super-replication Option Arbitrage No arbitrage opportunities Bellman–Isaacs equations Multivalued mapping Semicontinuity Continuity Modulus of continuity Lipschitz functions 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sergey N. Smirnov
    • 1
  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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