Does a forward price have a drift component in any measure?

Question

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.

Answer 1

I will formalize my answer later. But one thing you will have to know is that the price of a forward contract will be a martingale under t forward measure, meaning the drift term is 0. This is not true under other measures. So the drift of the process depends on the measure you use to price the contract.

Answer 2

It the value of the forward contract and not the forward price that has drift r under the risk-neutral measure. In fact, in the simplest case where the risk-free interest rate is a constant r, then the forward price process f(t,T) has zero drift under the risk-neutral measure: If the spot price process satisfies dS(t)=S(t)(rdt+bdW(t)), then dF(t,T)=bF(t,T)dW(t).