 Solution

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.