 Explicit log-price scheme for local-volatility pricing

With $x=\log S$ and $u=e^{r(T-t)}V(e^x,t)$, a local volatility pricing equation becomes the displayed backward equation. Central finite differences and the explicit Euler method give $u_i^{j-1}=u_i^j+k[a_i^j{D}_{xx}{u}_i^j+b_i^j{D}_x{u}_i^j]$. Nonnegative weights require $k{\sigma }_i^2/h^2\le 1$ and $|b_i|h\le {\sigma }_i^2$. Under these conditions the update is a convex combination, yielding maximum-norm stability. With smooth coefficients, positive diffusion, fixed positive remaining maturity and suitable consistent boundary treatment, its ordinary error is $O(k+h^2)$. Volatility bounds alone do not imply all this regularity or a Greek error expansion.