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Abstract <p>In the article, the European option superhedging problem is identified with a dynamic stochastic zero-sum game between the market and the contract seller. The seller manages a portfolio of underlying assets in order to minimize her expected exponential risk. The market determines a probability distribution for discounted prices of traded assets: absolutely continuous with respect to a given underlying distribution and maximazing seller’s expected risk. Recurrence relations for the upper and lower values of the game are obtained. It is shown that the absence of arbitrage opportunities in the market is a necessary and sufficient condition for the existence of a self-financing portfolio, with which the lower bound in the definition of the upper value of the game is obtained. Such a portfolio is superhedging, and the upper value of the game allows one to calculate the upper hedging price. Moreover, it is shown that, in a market model without arbitrage opportunities, there is always a game equilibrium. The saddle point of the game, if it exists, determines a superhedging portfolio and a martingale probability distribution, with which an upper bound in the definition of the upper value of the game is achieved. This distribution defines the seller’s worst market in the sense that the reserve of the superhedging portfolio is fully consumed in that market model. Using examples, we provide comparison between results of option calculations based on probabilistic and trajectory-based game approaches, analyze advantages and disadvantages of the topology choice (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma ({{L}^{1}},{{L}^{\infty }})\)</EquationSource> <!--DANMath2560101Shelemekh-m1--> </InlineEquation> topology instead of weak topology) in the probabilistic formulation of this type of problems.</p>

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European Option Superhedging as an Antagonistic Game

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • O. V. Zverev,
  • E. A. Shelemekh

摘要

Abstract

In the article, the European option superhedging problem is identified with a dynamic stochastic zero-sum game between the market and the contract seller. The seller manages a portfolio of underlying assets in order to minimize her expected exponential risk. The market determines a probability distribution for discounted prices of traded assets: absolutely continuous with respect to a given underlying distribution and maximazing seller’s expected risk. Recurrence relations for the upper and lower values of the game are obtained. It is shown that the absence of arbitrage opportunities in the market is a necessary and sufficient condition for the existence of a self-financing portfolio, with which the lower bound in the definition of the upper value of the game is obtained. Such a portfolio is superhedging, and the upper value of the game allows one to calculate the upper hedging price. Moreover, it is shown that, in a market model without arbitrage opportunities, there is always a game equilibrium. The saddle point of the game, if it exists, determines a superhedging portfolio and a martingale probability distribution, with which an upper bound in the definition of the upper value of the game is achieved. This distribution defines the seller’s worst market in the sense that the reserve of the superhedging portfolio is fully consumed in that market model. Using examples, we provide comparison between results of option calculations based on probabilistic and trajectory-based game approaches, analyze advantages and disadvantages of the topology choice ( \(\sigma ({{L}^{1}},{{L}^{\infty }})\) topology instead of weak topology) in the probabilistic formulation of this type of problems.