= Solution
Expand the update in part (b). Its <trinomial tree> weights are
$$
\boxed{p_{i,d}^j=\frac{k\widehat\sigma_i^{j,2}}{2h^2}-\frac{kb_i^j}{2h},\qquad
p_{i,m}^j=1-\frac{k\widehat\sigma_i^{j,2}}{h^2},\qquad
p_{i,u}^j=\frac{k\widehat\sigma_i^{j,2}}{2h^2}+\frac{kb_i^j}{2h}.}
$$
They connect the current state to $x_i-h,x_i,x_i+h$ at the next time. Their sum is one, their <mean> increment is $b_i^jk$, and their raw second increment moment is $\widehat\sigma_i^{j,2}k$. Thus they match the log-price diffusion to first order in the time step.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-48-trinomial.png]
{title=One-step log-price trinomial tree and its backward pricing average}
All three weights are probabilities precisely when they are nonnegative:
$$
\boxed{\frac{k\widehat\sigma_i^{j,2}}{h^2}\leq1,\qquad
|b_i^j|h\leq\widehat\sigma_i^{j,2},}
$$
at every node and level. They are then automatically at most one. With positive $\sigma_-$, a simple sufficient uniform choice is $k\leq h^2/\sigma_+^2$ and $h\sup|r-\widehat\sigma^2/2|\leq\sigma_-^2$. If volatility vanishes at a node while drift does not, the centered weights cannot all be nonnegative.
Nonnegative weights make the backward update a <convex combination>, proving maximum-norm <stability> and monotonicity: interior errors cannot exceed the larger of the terminal and boundary errors. This is a sufficient <stability> condition, not an equivalence with every possible <stability> notion. For constant coefficients on an unrestricted or periodic grid, let $\kappa=\widehat\sigma^2k/h^2$ and $\nu=bk/h$. The Fourier amplification factor is
$$
G(\vartheta)=1-2\kappa\sin^2(\vartheta/2)+i\nu\sin\vartheta.
$$
Requiring $|G|\leq1$ for every frequency gives $\kappa\leq1$ and $\nu^2\leq\kappa$. Indeed, with $s=\sin^2(\vartheta/2)$, $|G|^2-1=4s[(\nu^2-\kappa)+(\kappa^2-\nu^2)s]$, so its two endpoint inequalities suffice. Probability weights require the stronger $|\nu|\leq\kappa$. The distinction prevents interpreting a merely Fourier-stable scheme with negative weights as a probability tree.
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