# Dynamic spanning without probabilities

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## Abstract

The paper presents a non-probabilistic approach to continuous-time trading where, in analogy to the binomial option-pricing model, terminal payoffs resulting from a given trading strategy are meaningful 'state-by-state', i.e., path-by-path. In particular, we obtain results of the form: "If a certain trading strategy is applied and if the realized price trajectory satisfies a certain analytical property, then the terminal payoff is...." This way, derivation of the Black and Scholes formula and its extension become an exercise in the analysis of a certain class of real functions. While results of the above forms are of great interest if the analytical property in question is believed to be satisfied for almost all realized price trajectories (for example, if the price is believed to follow a certain stochastic process which has this property with probability 1), they are valid regardless of the stochastic process which presumably generates the possible price trajectories or the probability assigned to the set of all paths having this analytical property.

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