Derivation: Transforming Black-Scholes to the Heat Equation

The Black–Scholes PDE has variable coefficients ($S$, $S^2$) and runs backwards in time. The classical heat equation $y_\tau = y_{xx}$ has constant coefficients and runs forwards. This subnote walks through the substitutions that map one into the other, explains why each substitution is the natural one, and records the transformed payoffs and boundary data used by the finite-difference schemes of finite difference methods. The reward for the algebra: every piece of numerical analysis (stability, convergence, scheme design) can be done once, on the simplest parabolic PDE there is, then transported back to finance.