  Expanding the ideas of the author's paper 'Nonexpansive maps and option
pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure
game-theoretic approach to option pricing, by-passing stochastic modeling. Risk
neutral probabilities emerge automatically from the robust control evaluation.
This approach seems to be especially appealing for incomplete markets
encompassing extensive, so to say untamed, randomness, when the coexistence of
infinite number of risk neutral measures precludes one from unified pricing of
derivative securities. Our method is robust enough to be able to accommodate
various markets rules and settings including path dependent payoffs, American
options and transaction costs. On the other hand, it leads to rather simple
numerical algorithms. Continuous time limit is described by nonlinear and/or
fractional Black-Scholes type equations.
