The risk-neutral pricing formula is a central result of asset pricing theory that provides the proper valuation of European derivatives in absence of arbitrages. Despite the variety of option payoffs, price dynamics are driven by the same interest rate process. We formalize this intuition by showing that no arbitrage prices constitute the weak solution of a differential equation, where the risk-free rate has a prominent role. To achieve this goal, we introduce the notion of weak time-derivative, which permits to differentiate functions of Markov processes. It generalizes the infinitesimal generator and provides a characterization of martingales, submartingales and supermartingales; in particular, martingales are characterized by a null weak time-derivative. Through the weak time-derivative, we formulate the no arbitrage pricing differential equation and we prove existence and uniqueness of the solution with both deterministic and stochastic interest rates. Finally, we reformulate the eigenvalue-eigenvector problem of Hansen and Scheinkman (2009) and we retrieve a decomposition of the stochastic discount factor, well-known in the long-term risk literature.