= Finite-difference option Greeks
{title2=$\Delta=e^{-r(T-t)-x}u_x,\qquad\Gamma=e^{-r(T-t)-2x}(u_{xx}-u_x)$}
The logarithmic coordinate requires the displayed chain-rule factors for <option delta> and <option gamma>. If only a value error bound $\|e_h\|_\infty=O(h^2)$ is known, central differentiation gives $D_xe_h=O(h)$ 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 $e_h=h^2E+o(h^2)$ 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.
Back to article page
