= Solution
\b[Refining space alone at fixed positive time step does not generally converge.] The diffusion <Courant number> is $\widehat\sigma^2k/h^2$, 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 $u_i^{m-1}=ka_i^mK/h+O(1)$, unbounded as $h\downarrow0$ 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.
Back to article page
