Scientific publications

In the article, the European option superhedging problem is identified with a dynamic stochastic zero-sum game between the market and the seller of the contract. 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 respective to a given underlying distribution and maximazing seller's expected risk. The recurrence relations for the upper and lower values of the game are obtained. It is shown that 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 limit in definition of the upper value of the game is obtained. Such a portfolio is a superhedging one, and the upper value of the game allows you 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 exists, determines a superhedging portfolio and a martingale probability distribution, with which an upper bound in 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, analyse advantages and disadvantages of the topology choice σ(L¹;L^∞) topology instead of weak topology) in the probabilistic setting for a problems of the type.