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Pointwise arbitrage pricing theory in discrete time. (English) Zbl 1437.90159

Summary: We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain abstract (pointwise) fundamental theorem of asset pricing and pricing-hedging duality. Our results are general and, in particular, cover both the so-called model independent case as well as the classical probabilistic case of Dalang-Morton-Willinger. Our analysis is scenario-based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

MSC:

90C46 Optimality conditions and duality in mathematical programming
90C47 Minimax problems in mathematical programming
90C17 Robustness in mathematical programming
91G20 Derivative securities (option pricing, hedging, etc.)
49K45 Optimality conditions for problems involving randomness
49N15 Duality theory (optimization)
60G42 Martingales with discrete parameter
93E20 Optimal stochastic control
91G70 Statistical methods; risk measures
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References:

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