# 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

## 1 Answer

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.

For a derivation of the term structure from the Hull-White short rate model, see this answer.

• I would add the facts that popular model are designed to return to a 'long term average' and this long term average is calibrated using long time period. As interest rate are currently low and were higher previously, the calibrated long term average is generally higher than current interest rate levels. Overall popular model will genrally give fast growing future rate scenarios. – lcrmorin Aug 18 '16 at 14:58
• One factor short rate model models the instantaneous rate given any moment in time. Can you explain how is term structure derived from a short rate model ? – user1559897 Aug 18 '16 at 16:05
• 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
• So if we have the short rate model, cant we derive the forwards rate from? Then how is it different hjm where forward rates are directly modeled? – user1559897 Aug 20 '16 at 12:53
• @user1559897: From the short rate model, you can derived the bond price, and then, by taking derivatives with respect to the maturity of the log-bond price, you can obtain the forward rate. To directly model the forward as in HJM, you have more freedom for the volatility, however, the drift term takes a particular form related to the volatility function. – Gordon Aug 20 '16 at 13:14