An F-cone is a pointed and generating convex cone of a real vector space that is the union of a countable family of finite dimensional polyhedral convex cones such that each of which is an extremal subset of the subsequent one. In this paper, we study securities markets with countably many securities and arbitrary finite portfolio holdings. Moreover, we assume that each investor is constrained to have a non-negative end-of-period wealth. If, under the portfolio dominance order, the positive cone of the portfolio space is an F-cone, then Edgeworth allocations and non-trivial quasi-equilibria exist. This result extends the case where, as in Aliprantis et al. [J. Math. Econom. 30 (1998a) 347], the positive cone is a Yudin cone.