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Matching Initial Short Rates Across Vasicek and Nelson–Siegel Models

Article Quant Q&A · Author: Sentinel
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Summary

The document asks whether a Vasicek short rate can be set equal at the initial time to the short rate implied by a Nelson–Siegel curve. In the stated Nelson–Siegel specification, the initial short rate is the sum of the first two curve parameters, since the remaining term vanishes at zero maturity. The question relates this possible match to the initial zero-coupon bond price under Vasicek or Hull–White.

The answer says this equality can be a reasonable assumption when one needs to use a different short-rate model. However, independently calibrating each model does not guarantee matching initial rates: the Nelson–Siegel parameters may differ after separate fits. The exchange provides only this brief qualification and no calibration procedure, numerical example, or proof that matching the initial rate alone will align bond prices across maturities. It therefore highlights a modeling choice, not a general equivalence between the yield-curve and short-rate frameworks.

Key ideas

  • The Nelson–Siegel short rate at zero maturity equals the sum of its first two parameters.
  • A modeler may choose to match that initial rate to the Vasicek short rate.
  • Separate calibrations can produce different Nelson–Siegel parameters and therefore different initial rates.
  • Matching the initial short rate alone does not establish that the models produce the same full yield curve.

Tags

Fixed incomeStatistics
Full text
# Are instantaneous short rates compatible across models?


# 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: $P(0,T) = e^{A(0,T) -B(0,T) r_V(0)} \stackrel{?}{=} e^{A(0,T) -B(0,T) (β_0 + β_1)}$.

## Answer by THATS MY QUANT MY QUANTITATIVE (score 1)

https://quant.stackexchange.com/a/77345

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 $\beta_1$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.