The Binomial Price as a Risk-Neutral Expectation

Rewrite the price as $C_0 = \frac{1}{r}\left[p^*C_u + (1-p^*)C_d\right]$, show that the two weights sum to one, and record that $p^*$ is not the actual probability $p$.

State that the weights are non-negative exactly when $d < r < u$, record $C_0 \ge 0$, and describe what happens to bonds if $r < d$ and to the stock if $r > u$.

For each case $C_0 > V_0$ and $C_0 < V_0$, write the trades made today, the cash received, and how the positions cancel tomorrow, using one illustrative mispricing as the example.

Record that $S_0$ is the only market price in the model, while $C_u$ and $C_d$ are computed from the agreed values of $u$ and $d$.