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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.

The stock diffusion $d{S}_t=S_t(rdt+\sigma (t,S_t)d{W}_t)$ has spot-dependent local volatility. With positive volatility and the Brownian filtration, Brownian martingale representation theorem converts discounted payoff martingales into stock gains.

Let $\hat{\sigma }(x,t)=\sigma (e^x,t)$, $a(x,t)={\hat{\sigma }}^2/2$ and $b(x,t)=r-{\hat{\sigma }}^2/2$. The chain rule gives $S{V}_S=V_x$ and $S^2{V}_{SS}=V_{xx}-V_x$. Also $V=e^{-r(T-t)}u$, so $V_t=e^{-r(T-t)}(u_t+ru)$. The killing term cancels, leavingThis is the explicit log-price scheme for local-volatility pricing equation. Under the risk-neutral measure, the corresponding log-price Itô formula givesThe logarithmic coordinate removes the stock-price factors from the diffusion coefficient, but retains the spatial variation of local volatility.