Mathematical model of European option pricing in incomplete market without transaction costs (discrete time ). Part I
For European option in multidimensional incomplete market without transaction costs we design discreet time pricing model. At first the following auxiliary problem is to be considered: to find the upper guaranteed value for the expected risk depending exponentialy on a shortage. The upper guaranteed value is a minimax of the expected risk. First we take supremum over a set of equivalent probability measures. Then we take infimum over a set of self-financing portfolios. Here we find conditions for the existence of a portfolio such that an infimum is attained. We use this result to find a generalized optional decomposition for a contingent claim. Further, we obtain conditions for the existence of a probability measure such that the expected risk is maximal with respect to the measure. This measure turned out to be martingale and discreet and it does not belong to the set of equivalent measures. Finally, we demonstrate that our auxiliary results make it possible to obtain explicit pricing formulas for an European option in an incomplete market without transaction costs. In part II of the paper we present example models of European options’ pricing in a one-dimensional market and in a market, where support of basic probability measure is compact.