# 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

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.

• Thanks for your answer. I'd like to ask about two more things just to have a clear view: 1) In practice, what are the usual instruments you calibrate short-rate models to? Vasicek can be also calibrated to historical data- which of the approaches is more popular in practice? 2) Could you suggest any models/classess of models that are capable of modelling LIBOR for different tenors simultaneously? – Bart D. May 6 '16 at 6:54
• The usual practice (in the US) is to calibrate all models to the 3 month libor curve, specifically to the prices of Eurodollar futures and interest rate swaps. The 1 month libor curve and the 6 month libor curve would then be derived by applying a deterministic spread to the 3 month curve. – dm63 May 7 '16 at 14:21
• Ok, makes sense. Could you please describe the calibration to the yield curve in greater detail or direct me to a specified procedure described on the internet? Up until today, I have only calibrated Vasicek via regression to historical realizations of libor (i.e. using this procedure: sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model) – Bart D. May 7 '16 at 17:41
• I'm not sure I have the answer you need. In practice no-one uses historical data or least squares analysis to calibrate interest rate models. That's because models must exactly reprice all visible swap rates, otherwise you get immediately arbitraged. Least squares fits are not good enough. This means that a model with 2 or 3 parameters like the basic Vasicek is not rich enough to be useful, so therefore I have no experience of the calibration you seek. Apologies – dm63 May 7 '16 at 22:33
• Sure thing, I was fond of Vasicek due to its simplicity and the fact that I do not need to value any instruments- I merely want to have any projection of future interest rates based on today's state of the world. What classes of models do you use personally that address your concerns? – Bart D. May 8 '16 at 15:48

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.