Revista de la
Unión Matemática Argentina
Conditional non-lattice integration, pricing, and superhedging
Christian Bender, Sebastian E. Ferrando, and Alfredo L. Gonzalez

Volume 68, no. 2 (2025), pp. 627–676    

Published online (final version): October 8, 2025

https://doi.org/10.33044/revuma.4351

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Abstract

Motivated by financial considerations, we develop a non-classical integration theory that is not necessarily associated with a measure. The base space consists of stock price trajectories and embodies a natural no-arbitrage condition. Conditional integrals are introduced, representing the investment required to hedge an option payoff when entering the market at any later time. Here, the investment may depend on the stock price history, and hedging takes place almost everywhere and as a limit over an increasing number of portfolios. In our setting, the space of elementary integrands fails to satisfy the lattice property and the notion of null sets is financially motivated and not measure- theoretic. Therefore, option prices arise from conditional non-lattice integrals rather than expectations, with no need to impose measurability assumptions.

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