= Solution
In the one-period building block of the <Cox--Ross--Rubinstein model>, the money-market account grows by a deterministic factor $R>0$, while the stock grows independently at each time by
$$
Y_t=\begin{cases}U,&\text{up},\\D,&\text{down},\end{cases}
\qquad 0<D<U.
$$
Thus $S_t^0=R^tS_0^0$ and $S_t^1=S_0^1\prod_{j=1}^tY_j$. Equivalently, the discounted stock has multipliers $u=U/R$ and $d=D/R$.
The model has no <arbitrage> precisely when the risk-free return lies strictly between the stock returns:
$$
\boxed{D<R<U,\qquad\text{equivalently}\qquad d<1<u.}
$$
Indeed, failure of either strict inequality gives a one-period arbitrage, while under the strict inequalities the probability
$$
\boxed{q=Q(Y_t=U)=\frac{R-D}{U-D}=\frac{1-d}{u-d}}
$$
lies in $(0,1)$. Taking the moves independent under $Q$ with up probability $q$ makes the discounted stock a <martingale>, since $qu+(1-q)d=1$. This is the unique <Risk-neutral probability in the Cox--Ross--Rubinstein model>, hence the unique equivalent martingale measure.
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