Why One-Factor Affine Short-Rate Models Shift Yields in Parallel

Summary

The document shows how an affine short-rate model maps the current short rate into zero-coupon bond prices and yields. Bond prices have an exponential-affine form: deterministic maturity-dependent terms multiply an exponential involving the current short rate. Taking the negative log price per unit maturity gives a yield that is an affine function of that same rate.

Because yields at different maturities depend on a single stochastic factor, their correlation at a fixed time is perfect under the stated affine setup. A change in that factor therefore moves yields together, producing a parallel shift in the curve in this simplified sense. The explanation gives an algebraic argument rather than a calibration or empirical test. It does not imply that observed yield curves always shift in parallel; models with additional factors can represent changes in slope or curvature.

Key ideas

• Affine short-rate models express zero-coupon bond prices using deterministic maturity functions and the current short rate.
• The resulting yield at each maturity is an affine function of the same short-rate factor.
• Yields across maturities are perfectly correlated at a fixed time in the stated one-factor setup.
• A change in the single factor therefore produces a parallel yield shift within this model structure.
• The conclusion is a simplifying model property, not a claim about all observed yield movements.

Tags

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


# 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?

## Answer by Gordon (score 9, accepted)

https://quant.stackexchange.com/a/29714

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.

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