= Solution
{c}
Write $R=e^r$, $u=4/3$ and $d=2/3$. Since both moves have positive physical <probability>, the <binomial options pricing model> is free of <arbitrage> exactly when
$$
\boxed{\frac23<e^r<\frac43.}
$$
If $R\leq d$, borrowing the share price and buying one share gives terminal gain $S_t(u-R)$ or $S_t(d-R)$, both nonnegative and at least one positive. If $R\geq u$, shorting one share and investing its proceeds gives the opposite comparison, with gain $S_t(R-u)$ or $S_t(R-d)$. Equality at either endpoint still permits <arbitrage> because the other outcome has positive <probability>.
Conversely, strict inequalities give a <risk-neutral probability> $q=(R-d)/(u-d)\in(0,1)$, for which the discounted share is a <martingale>. Any zero-cost <self-financing portfolio> with nonnegative terminal value has zero discounted <expectation> under this equivalent positive-probability law and hence cannot have a strictly positive payoff in either state. Applying this argument conditionally at each node proves <no-arbitrage> over the tree.
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