Explicit scheme for the heat equation and its stability

After transforming the pricing PDE into the heat equation $u_\tau=u_{xx}$, show how an explicit finite-difference scheme advances the solution, and state the condition under which it is stable.

$u_j^{n+1}=\alpha u_{j-1}^n+(1-2\alpha)u_j^n+\alpha u_{j+1}^n$ with $\alpha=\Delta\tau/(\Delta x)^2$; stable iff $\alpha\le\tfrac12$. Use a forward difference in $\tau$ and a central second difference in $x$. Von Neumann analysis (or requiring nonnegative weights) shows errors are damped only when $\alpha\le\tfrac12$, so halving $\Delta x$ forces quartering $\Delta\tau$.