A financial market on a stochastic basis with a filtration generated by the binary tree is considered. A plugin simulating failures on this market is constructed. We mean by a failure the following situation on the financial market: when passing from a time moment  to the next one new events arise but discounted price of the fixed type stock does not change. A failure generates the incompleteness of the market (the set of martingale measures of this market is infinite). By modelling of a weak deformation it is possible to reduce the set of martingale measures to a unique measure. Thereby the price of every contingent claim is uniquely determined. This price may be considered as "fair price".

Keywords: Stochastic basis, probability measure, financial market, binary tree, arbitrage free, completeness, weak deformation, martingale measure, plugin.

With the help of so-called deformations (that is of probability measures families on sigma-fields forming a filtration) a notion of deformed martingale is introduced. This notion generalizes the classical concept of martingale with discrete time. We differ two sorts of deformed martingales: deformed martingales of the 1-st and the 2-nd type. Similarly deformed sub- and supermartingales of the 1-st and the 2-nd type are introduced. We prove that infimum of arbitrary family of deformed supermartingales is a supermartingale and that convexe function of deformed martingale is a deformed submartingale. In addition, for deformed martingales of the 2-nd type a telescopic property is obtained.

Keywords: Filtration, probability measure, deformation, deformed martingale, telescopic property