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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.

Refining space alone at fixed positive time step does not generally converge. The diffusion Courant number is ${\hat{\sigma }}^{2}k/h^2$, which grows by four at each halving of $h$. For positive volatility it eventually violates the explicit diffusion stability restriction. The explicit log-price scheme for local-volatility pricing then amplifies grid-scale errors, rather than approaching the pricing solution.

The strike kink makes the failure concrete. At the terminal node $x_i=\log K$, $u_i^m=0$, while $D_{xx}{u}_i^m=K/h+O(1)$. Even a single step of fixed size $k$ gives $u_i^{m-1}=k{a}_i^{m}K/h+O(1)$, unbounded as $h\downarrow 0$ if $a_i^m>0$. Thus the problem is not cured by monitoring exactly at the money. A stable joint space-time refinement is required; the zero-diffusion case does not support the same diffusion argument and centered explicit advection has its own stability problem.

A trinomial tree allows three next-step states. For the explicit log-price scheme for local-volatility pricing these are $x-h,x,x+h$, with the displayed weights. They sum to one; if nonnegative they are transition probabilities and make backward pricing a conditional average. Their first two increment moments are $bk$ and $2ak$, matching the diffusion to first order in the time step. Coefficients can depend on the node, so the tree need not have constant transition probabilities.