# Why does one-factor short-rate model tend to produce parallel shift of the yield curve?

I understand that one factor short rate model models the instantaneous rate given any moment in time. Can anyone explain how to derive a term structure from a short rate model and show that one-factor short-rate model tend to produce parallel shift of the yield curve?

• A model itself does not produce a parallel shift. Are you talking about certain risk? The duration is based on parallel shift to the yield. A short rate model produce bond prices with perfectly correlated increments. – Gordon Aug 18 '16 at 1:16
• I guess a change of certain parameters produces parallel shift of yield curve? – user1559897 Aug 18 '16 at 1:18
• You may need to modify your question to make it clear; otherwise, we do not know what you are asking. – Gordon Aug 18 '16 at 1:19
• @Gordon He's asking how to prove perfect correlation for the curve points in Vasicek model. I don't know to prove it but I know that's because the model doesn't have enough parameters to address the issue. – SmallChess Aug 18 '16 at 6:18
• @Gordon Aren't they the same thing? Zero rates should be perfectly correlated with bond price? They are just different representation of the same data. – SmallChess Aug 18 '16 at 13:20

This has already been explained at the start of Chapter 4 in Brigo's book. Basically, for any affine model of the short rate $r_t$, the zero-coupon bond price has the form \begin{align*} P(t, T) = A(t, T)e^{-B(t, T) r_t}, \end{align*} where $A(t, T)$ and $B(t, T)$ are deterministic functions. The yield, or zero rate, is given by \begin{align*} R(t, T) &= -\frac{\ln P(t, T)}{T-t}\\ &=-\frac{\ln A(t, T)}{T-t} + \frac{B(t, T)}{T-t} r_t\\ &=:a(t, T) + b(t, T) r_t. \end{align*} Then \begin{align*} {\rm Corr}\big(R(t, T_1), R(t, T_2) \big) &= {\rm Corr}\big(a(t, T_1) + b(t, T_1 r_t, a(t, T_2) + b(t, T_2) r_t \big)\\ &={\rm Corr}(r_t, r_t) =1. \end{align*} That is, at any time $t$, the yield to any two maturity dates are perfectly correlated, and any shift to a single yield causes a parallel shift to the whole yield curve.
• A term structure, at time $t$, can mean either the yield $R(t, T)$ or the bond price $P(t, T)$ for a sequence of maturity dates $T$. An example of such derivation is pointed out in the last line of the answer. – Gordon Aug 18 '16 at 16:17