 Finite-difference option Greeks

The logarithmic coordinate requires the displayed chain-rule factors for option delta and option gamma. If only a value error bound $\Vert e_h{\Vert }_{\infty }=O(h^2)$ is known, central differentiation gives $D_x{e}_h=O(h)$ and $D_{xx}{e}_h=O(1)$. Thus first and second Greek convergence do not follow with the same order from a value bound alone. With a smooth spatial error expansion $e_h=h^{2}E+o(h^2)$ and corresponding derivative control, both Greeks can instead retain second-order convergence away from a nonsmooth payoff or boundary. Refinement and Greek-specific regularity are essential to distinguish these situations.