We consider the maximal extension (excluding jump processes) of the one-factor Black-Scholes-Merton option pricing model. We argue that this model and the local volatility approach provide the optimal one-factor option pricing model. We present a formalism for determining sensitivities to variations in volatility, interest rate or dividend rate when these quantities are functions of any or all of the independent variables. This formalism results in partial differential equations for the sensitivities which have gamma or delta (depending on the particular case) as source terms. We show that these new expressions for the sensitivities reduce to the usual partial derivatives in the case where the relevant parameter is constant. We include various numerical examples to illustrate these ideas.