The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space, trading is assumed to take place at a finite number of occasions but not bounded in number nor necessarily equally spaced in time. For a given option, there exists an interval bounding the set of possible fair prices; such interval exists under more general conditions than the usual no-arbitrage requirement. The paper develops a backward recursive method to evaluate the option bounds; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. Trajectory sets are introduced for which existing non-probabilistic markets models are nested as a particular case. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.

ABSTRACT. The paper develops general, discrete, non-probabilistic market models and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of martingalelike properties as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model.

H-systems are those orthonormal systems which allow computation of conditional expectations via a Fourier expansion. These systems provide natural approximations to continuous stochastic processes and we indicate how they could be used to perform lossy compression on a set of given signals. More specifically, we show how to construct, for a given random variable, an adaptive martingale approximation by means of generalized Haar functions. We also indicate how the construction can be extended to a given random vector. A generalized multiresolution analysis algorithm is also described and numerical examples are provided.

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The paper develops general, non-probabilistic market models based on trajectory sets and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of a martingale process as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. An illustrative example is described in detail. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model.