# Are instantaneous short rates compatible across models?

If I calibrate the Vasicek's yield curve to the Nelson-Siegel's (NS) yield curve, can I assume that $$r_V(0) = r_{NS}(0) = \beta_0 + \beta_1$$ or not?

NS short rate:

$$r_{NS}(S) = β_0 + β_1 e^{-S/\tau} + β_2 \frac{S}{\tau} e^{-S/\tau}$$

$$r_{NS}(0) = β_0 + β_1$$

The today's zero coupon bond price under Vasicek and Hull-White models is: $$P(0,T) = e^{A(0,T) -B(0,T) r_V(0)} \stackrel{?}{=} e^{A(0,T) -B(0,T) (β_0 + β_1)}$$.

• If I understand correctly: In other words: you have a historical time series of short rates $r(t_0),r(t_1),\dots \,.$ Before deciding which models them best it would be a good idea to look at a graph. If they never become negative the normally distributed HW model might not be the best choice. Since no model is the true one for any time series there is no proof which model is the right one. Hence I don't see any grave errors. Only better or worse choices. Nov 11, 2023 at 11:17

It can be a reasonable assumption if for whatever reason you needed to use a different model for the instantaneous short rate. But if you calibrated them separately, you would get different $$\beta_0$$ and $$\beta_1$$.