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For smooth derivative prices depending on a Markov short rate, Itô formula gives physical drift $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and exposure $s=\sigma V_r$. The common short-rate market price of risk gives $m-rV=\theta s$, yielding this equation. Under the risk-neutral measure, the rate drift is $b-\sigma \theta$ and the bank-discounted price is a martingale under suitable integrability. A unit zero-coupon bond has terminal value one and price ${\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$.

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 $-s_2{V}_1+s_1{V}_2$ in the bank account. The portfolio has zero current value and zero diffusion exposure. Its instantaneous excess drift isIf 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 exposuresAt 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 $\theta$.