Summary: We study the convexity property of the set \(Q_{\mathcal{F}}\) of arbitrage-free prices of a multi-period financial structure \(\mathcal{F}\). The set of arbitrage-free prices is shown to be a convex cone under conditions on the financial structure \(\mathcal{F}\) that hold in particular for short-lived assets. Furthermore, we provide examples of equivalent financial structures \(\mathcal{F}\) and \(\mathcal{F}'\) such that \(Q_{\mathcal{F}}\) is a convex cone, but \(Q_{\mathcal{F}'}\) is neither convex nor a cone.