I’ve introduced the Vasicek stochastic rates model in an earlier post, and here I’m going to introduce a development of it called the Hull-White (or sometimes Hull-White extended Vasicek) model.

The rates are modelled by a mean-reverting stochastic process

which is similar to the Vasicek model, except that the term is now allowed to vary with time (in general and are too, but I’ll ignore those issues for today).

The freedom to set theta as a deterministic function of time allows us to calibrate the model to match the initial discount curve, which means at least initially our model will give the right price for things like FRAs. Calibrating models to match market prices is one of the main things quants do. Of course, the market doesn’t really obey our model. this means that, in time, the market prices and the prices predicted by our model will drift apart, and the model will need to be re-calibrated. But the better a model captures the various risk factors in the market, the less often this procedure will be needed.

Using the trick

to re-express the equation and integrating gives

where is the rate at . The observable quantities from the discount curve are the initial discount factors (or equivalently the initial forward rates) , where

The rate is normally distributed, so the integral must be too. This is because an integral is essentially a sum, and a sum of normal distributions is also normally distributed. Applying the Ito isometry as discussed before, the expectation of this variable will come wholly from the deterministic terms and the variance will come entirly from the stochastic terms, giving

where throughout

and since

we have

Two differentiations of this expression give

and combining these equations gives an expression for that exactly fits the initial discount curve for the given currency

and since is simply the initial market observed forward rate to each time horizon coming from the discount curve, this can be compactly expressed as

Today we’ve seen how a simple extension to the ‘basic’ Vasicek model allows us to match the initial discount curve seen in the market. Allowing the volatility parameter to vary will allow us to match market prices of other products such as swaptions (an option to enter into a swap), which I’ll discuss another time. But we’re gradually building up a suite of simple models that we can combine later to model much more complicated environments.

Regarding the last para, any chance of showing us how to fit swaption prices by introducing a time-dependence to the volatility parameter?

Great idea – I’ll make this my next post!