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I don't know exactly about Reuters but often implied volas in the Black 76 world are quoted (forward) ATM. Thus the forward equals the strike and they dissappear from the formula: $$ C = E[(F-K)^+] = \exp(- r t) (F N(d_1) - K N(d_2)) $$ and $d_1 = (\log(F/K)+\sigma^2/2T)/(\sigma \sqrt{T})$ and $d_2 = d_1 - \sigma \sqrt{T}$, see e.g. here. In the ATM case ...


If we are going to have the form \begin{align*} dr = A dt + BdW_t, \end{align*} Then both A and B are functions of $t$ and $r_t$, otherwise, $r_t$ is normal. However, note that \begin{align*} r_t = \exp\Bigg(\frac{1}{\sigma(t)}\bigg(\int_0^t \theta(s)\sigma(s) ds +\sigma(0)\ln r_0 + \int_0^t\sigma^2(s) dW_s\bigg)\Bigg). \end{align*} That is, $r_t$ is ...

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