The Black-Scholes PDE can be used to price any European contingent claim with a payoff that is only dependent on the underlying's price at maturity, for instance forwards and vanilla options. In the case of forwards, the PDE is not required, c.f. the cash-and-carry technique. Is it possible to show that a delta of a European derivative is equal to one if and only if the payoff is linear in stock price, i.e. in the form $f(S) = S - K$?
I might be wrong about this, but I think any derivative with linear payoff is just a forward (even in the case where the coefficient on $S$ is not one, the derivative is a portfolio of forwards plus/minus cash). I'm basically trying to justify why one needs a model for the stock in order to price an option but not a forward, i.e. is it the non-negativity or the non-linearity of the payoff that makes pricing hard?