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Trajectorial asset models with operational assumptions

1 Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, Ontario M5B 2K3, Canada
2 Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada
3 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, Mar del Plata 7600, Argentina

The paper addresses the problem of providing a framework and an algorithm to evaluate super and sub replicating prices, for European options, having interesting risk-reward characteristics. A general operational framework is put forward and illustrated by an algorithmic construction of one-dimensional models for option pricing. Asset models are defined based on a class of investors characterized by how they operate on financial data leading to potential portfolio rebalances. Once observable variables are selected for modeling, necessary conditions constraining these variables and resulting from the operational setup are derived. Future uncertainty is then reflected in the construction of combinatorial trajectory spaces satisfying such constraints. As the risky asset unfolds, it can be tested dynamically for the validity of observable sufficient conditions that rigorously imply the validity of the models. The paper describes the resulting algorithmic construction of such trajectory spaces and, in the absence of probability assumptions, a minmax algorithm that is available to evaluate the super and sub replicating prices.
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