= Solution
Write $q_{15},q_{12},q_9$ for the state probabilities under an <equivalent martingale measure>. Since cash is constant and the <European put option> pays only in the lowest state, the pricing equations are
$$
q_{15}+q_{12}+q_9=1,\qquad15q_{15}+12q_{12}+9q_9=10,\qquad2q_9=\xi_0.
$$
Their unique solution is $q_9=\xi_0/2$, $q_{15}=\xi_0/2-2/3$, $q_{12}=5/3-\xi_0$. Every physical state has positive <probability>, so equivalence requires all three values to be strictly positive. Thus
$$
\boxed{\frac43<\xi_0<\frac53.}
$$
For each price in this open interval the <pricing kernel> takes values $3q_{15},3q_{12},3q_9$ in the three equally likely states, proving absence of <arbitrage>.
The necessity, including the exclusion of endpoints, can also be checked directly. The three terminal payoff vectors of cash, the <stock> and the <European put option> form the invertible matrix
$$
\begin{pmatrix}1&15&0\\1&12&0\\1&9&2\end{pmatrix},\qquad\det=-6.
$$
Each unit state payoff therefore has a <replicating strategy>, whose initial cost is the corresponding $q$. A zero or negative $q$ supplies an <arbitrage>, so the endpoints are genuinely excluded.
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