= Solution
The <option delta> is a derivative in $S$, not in $x$. With $d=e^{-r(T-t)}$, the logarithmic chain rule gives
$$
\Delta=d e^{-x}u_x,\qquad \Gamma=d e^{-2x}(u_{xx}-u_x).
$$
Thus use $\Delta_i=d e^{-x_i}D_xu_i$ for the <delta hedge>, and $\Gamma_i=d e^{-2x_i}(D_{xx}u_i-D_xu_i)$ for <option gamma>. These formulas also identify the <finite-difference option Greeks> correctly on the equally spaced log-price grid.
There are two distinct accuracy statements. A smooth exact function has central first- and second-derivative truncation errors of order $h^2$. But the computed values themselves have error $e_i=O(h^2)$ under the coupled refinement. Without further control on its spatial variation,
$$
|D_xe_i|\leq\frac{|e_{i+1}|+|e_{i-1}|}{2h}=O(h),\qquad
|D_{xx}e_i|\leq\frac{|e_{i+1}|+2|e_i|+|e_{i-1}|}{h^2}=O(1).
$$
Therefore \b[the value error bound alone guarantees only first-order delta accuracy and does not guarantee convergent gamma]. For example grid error $e_i=h^2\sin(\pi i/2)$ has a central first difference of order $h$ and a second difference of order one, despite second-order value accuracy.
If instead the scheme has a smooth error expansion $e_i=h^2E(x_i,t)+o(h^2)$ with corresponding derivative bounds, both Greeks can retain second-order accuracy at smooth interior points. Hence reliable gamma is possible with additional regularity and a verified Greek refinement, but cannot be inferred merely from the value convergence in part (e). This qualification is important because the stated bounded-volatility assumption supplies no such error-derivative control.
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