We will take \(G(P(t),t)\) to figure the price of an option, with \(t\) being time, strike price \(X\) (not introduced yet), and expiration date \(T > t\) on a stock with price \(P(t)\) at time \(t\).
This representation does something really important: it expresses \(G\) as a function of only the current stock price \(P(t)\).