2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210132In this paper, we introduce a numeraire-free and original probability based framework for financial markets. We reformulate or characterize fair markets, the optional decomposition theorem, superhedging, attainable claims and complete markets in terms of martingale deflators, present a recent result of Kramkov and Schachermayer (1999, 2001) on portfolio optimization and give a review of utility-based approach to contingent claim pricing in incomplete markets.ProbabilityComputational Finance60H30, 60G44text