If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$
and want to find the short/spot rate $r_t$, is this simply:
$$f(t,t) = f(0,t) + \sigma^2t(t-\frac{t}{2}) + \sigma B_t$$
Can anything further be done?