# Why is the mean time-dependent in the Hull-White interest rate model?

In the Vasicek interest-rate model, the interest rate reverts to a constant mean. This makes sense to me. In my conception, the mean ought to be time-invariant, since interest rates don't follow an increasing or decreasing trend in the long term.

In the Hull-White modified model, the mean is a time-dependent function. I cannot understand why this is the case.

• Since when are interest rates insulated from time-varying changes? tradingeconomics.com/united-states/interest-rate (chose the start date to be 1971, long-term enough?) – Matthias Wolf Aug 2 '13 at 2:46
• @MattWolf Thanks for the link. I didn't mean to suggest that rates don't vary with time, but that the rate mean doesn't vary with time. For example, in this chart, rates have an average around 5% – tjahrenholz Aug 2 '13 at 21:11
• the choice of model always should be determined by the specific use case. If you model product with a duration of 30-50 years then sure you are most likely right to assume a long-term mean of around 5%. But if you look to model interest rate products with, let's say, 2 years of maturity then you should hardly plug in a 5% rate in this current environment. Hence some models assume a time varying mean. – Matthias Wolf Aug 3 '13 at 2:24

The claim that interest rates don't follow long term trends is not consistent with observed data. The idea of mean reversion is that interest rates do not rise or fall without bound, but are limited by economic and political factors. But there is no indication that this oscillation of short rates should happen around a constant mean. Allowing the mean reversion parameters to be time-dependent (as the Hull-White model does) allows the short rate (which is described by the model) to match the term structure of interest rates (forward rates).

• Thanks for your help. I'm still a little confused. I'll try to lay it out a little better. In order to model rates, we define a process whereby rates oscillate around a mean. However, if that mean is not constant, then we have to model it as well, which brings us back to our first problem- how do we model rates? – tjahrenholz Aug 2 '13 at 21:13
• As I mentioned in my answer, by adjusting the parameters (mean reversion rate and level) of the model in such a way that it fits the term structure. – FQuant Aug 2 '13 at 22:40

In Equity or FX, you are modelling the dynamic of a single number $S_t$ the stock price or fx spot rate.

In interest rate, you are modelling the dynamic of a curve $(f(t,t+\theta))$ all the forward rates (equivalently all the discount factors) for all tenors $\theta$.

So mean reversion, in the context of rates should be mean reversion around a mean curve. This mean curve is the forward curve built from the prices of deposits, futures and swaps observed in the market.

In the Vasiceck model the short rate mean reverts to a long term mean $r_\infty$. This means that you are mean reverting to a flat curve. But curves observed in the market are never flat and the model cannot even fit the forward curve observed at time 0. The Hull-White model improves on this by allowing you to fit the current expectation of the rates curve and mean revert to it. This requires replacing two of your parameters (the initial and long term short rates) by a whole curve (the initial forward curve which appears in the model in the form of the tenor dependent mean $\varphi(t)$).

Hope this helps