Short-Rate Models, LIBOR Curves, and Multicurve Basis
Summary
The document considers whether simulated instantaneous rates from a Vasicek or Hull–White model can be integrated to produce observed one- or three-month LIBOR. Its answers explain that a one-factor Hull–White setup is typically calibrated to one market curve, often the discount curve, and fitted to initial market bond prices using a time-dependent adjustment to the short rate.
In a multicurve framework, LIBOR projection curves differ from the discount curve, with basis swaps reflecting that difference. A single-curve short-rate model therefore cannot generally match both observed LIBOR tenors at once. The discussion describes a constant-spread multicurve adjustment as one practical simplification, and cautions that instantaneous rates are tied to a specified curve. It offers conceptual guidance rather than a detailed calibration procedure; the assumption of a constant spread is a modeling simplification, not a general description of curve dynamics.
Key ideas
- A short-rate model is associated with a particular rate curve.
- Hull–White’s time-dependent adjustment can fit the model to initial market bond prices.
- A one-factor model calibrated to one curve cannot generally fit multiple LIBOR curves simultaneously.
- Basis swaps capture differences between LIBOR projection and discount curves.
- A constant multicurve spread is a simplifying adjustment rather than a full account of curve behavior.
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Full text
# LIBOR rates from Vasicek/Hull-White model?
# LIBOR rates from Vasicek/Hull-White model?
I am somehow puzzled by the following problem: LIBOR rates are forward rates for an interbank loan for 1M or 3M (let's limit the range of possibilities to these two cases). Assuming that I have estimated the parameters of any short-term model (Vasicek, Hull-White etc.) and simulate the paths of instaneous rates, can I model market-observed LIBOR 3M as integral of instaneous rates over 3M span and similarly LIBOR 1M as integral over 1 month of instaneous rates? Or there is no link between market-observed LIBOR rates and instaneous rate that is modelled in the short-rate framework. Help me out!
Regards, Bart
## Answer by dm63 (score 4, accepted)
https://quant.stackexchange.com/a/25858
In practice, you can calibrate to either 1 month libor or 3 month libor, but not both. That's because there's a basis swap between 1 month libor and 3 month libor that can't be explained by your model.
## Answer by Nikita Kapitan (score 2)
https://quant.stackexchange.com/a/54757
In practice, 1-factor Hull-White model assumes the short rate to be:
$r_{t}=X_{t}+\varphi(t)+f^{M}(0, t)$
where
$X_t$ is pure mean reverting process: $ \mathrm{d} \mathrm{X}_{\mathrm{t}}=-\mathrm{a} \mathrm{X}_{\mathrm{t}} \mathrm{dt}+\sigma(\mathrm{t}) \mathrm{d} W_{\mathrm{t}}$
$f^M(0,t)$ is a market observed forward rate $\mathrm{f}^{M}(0, \mathrm{t})=-\frac{\partial}{\partial \mathrm{T}} \ln \mathrm{P}^{\mathrm{M}}(0, \mathrm{T})$
and $\varphi(\mathrm{t})=\int_{0}^{\mathrm{t}} \sigma^{2}(\mathrm{s}) \mathrm{e}^{-\mathrm{a}(\mathrm{t}-\mathrm{s})} \frac{1-\mathrm{e}^{-\mathrm{a}(\mathrm{t}-\mathrm{s})}}{\mathrm{a}} \mathrm{d} s$ is derived term that allows us to match the market bond prices:
So that we always have $P^{Market}(0, T)=\mathbb{E}\left[e^{-\int_{0}^{\top} r_{u} d u}\right]$
Answering your question, as you can see, our process is built on one and only one rate curve (normally discount curve) so that we match the bond prices (money market).
However today, in multicurve framework, where the LIBOR estimation curve is no longer equal to discounting curve, it's not possible to match the market-observed LIBOR rates with 1-factor Hull-White model.
The solution is to apply so called multicurve adjustments that is defined as:
- today's difference between discount and LIBOR estimation curve.
In this case we assume that the multicurve spread is constant.
Note, that instantaneous rate is just an object related to some rate curve.
You can have instantaneous rates for discounting curve as well as for LIBOR1M or LIBOR3M.
But instantaneous rates for LIBOR curves have no sense.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.