Let $\hat{k}^{x,y}_\lambda$ denote the numerical solution produced by the finite difference scheme on the dyadic grid $\mathcal{D}^\lambda$. Then \begin{align*} \sup_{(s,t) \in [a,b]^2} \left| k^{x,y}(s,t) - \hat{k}^{x,y}_\lambda(s,t) \right| \lesssim \frac{1}{2^{2\lambda}}, \end{align*} uniformly for all $\lambda \geq 0$, where the implicit constant depends on the paths $x$ and $y$ but not on $\lambda$.