 Solution

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.