The Black-Scholes equation for the price u(x,t) of a call option for a single share of common stock with dividend policy d(x,t) is 12σ2x2uxx+(rx−d(x,t))ux−ru−ut=0, 0 extlessx, 0 extlesst extlessT, with boundary conditions u(x,0)=max(0,x−E), 0≤x, u(0,t)=0, 0≤t≤T. The coefficients are unbounded and the equation is not uniformly parabolic. We prove an existence (and recall a uniqueness) theorem for a class of equations with boundary conditions that includes the Black-Scholes equation. These may be used to show that an American option must, or will not, sell for the same price as a European option.