There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this paper we establish such a converse. We find a Feynman–Kac-type theorem showing that the stochastic representation yields a classical solution to the corresponding Black–Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case.