Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward price process comes solely from the underlying price process. That much I can intuitively accept. But what about the drift component of the forward price process? Is it non-zero and does the choice of measure (risk-neutral, forward, real-world) influence its presence and magnitude?
Another issue I cannot reconcile is that I've read that all tradable securities under the risk-neutral measure must have a drift component the same as the risk-free interest rate. A forward contract is surely a tradable security?
Please excuse my confusion and possible bludgeoning of different concepts.