= Solution
The common <short-rate market price of risk> follows from absence of <arbitrage>, with the securities driven by the same <Brownian motion>. Take two traded derivatives $V_1,V_2$ with drifts $m_1,m_2$ and diffusion coefficients $s_1,s_2$. At a given state hold $s_2$ units of the first and $-s_1$ of the second, and place $-s_2V_1+s_1V_2$ in the <bank account>. The portfolio has zero current value and zero diffusion exposure. Its instantaneous excess drift is
$$
s_2(m_1-rV_1)-s_1(m_2-rV_2).
$$
If this were nonzero, selecting its positive orientation and maintaining the local risk-cancelling <self-financing> hedge would generate a gain without risk from zero wealth. Under the usual trading and regularity assumptions this violates absence of <arbitrage>. Thus the expression vanishes, and at nonzero exposures
$$
\boxed{\frac{m_1-rV_1}{s_1}=\frac{m_2-rV_2}{s_2}=\theta(t,r).}
$$
At zero exposure the correct relation is $m-rV=\theta s=0$, without dividing by zero. This is a condition on traded securities: the physical drift of the nontraded <short rate> alone does not determine $\theta$.
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