# Volatility in short-rate models and vol practical issues

I am slightly confused about the volatility term when pricing zero coupon bonds in the Ho-Lee model (and generally about where to get vol from in these kind of short rate models).

A particular framework could be the following. We start with the original non-risk neutral dynamics of the short rate specified by $$dr_t = \theta\,dt + \sigma\,dW_t$$ However, under the equivalent martingale probability measure, it can be proved that our zero-coupon bond price dynamics are given by $$dP(t, T) = r_t\,P(t, T)\,dt - (T-t)\,\sigma\,P(t,T)\,d\hat{W}_t$$

I am trying to find an answer to the following two questions:

1. Given that the $\sigma$ above is the vol of the short rate today ($t = 0$), can we use that same figure to price $P(0, T_1)$ and $P(0, T_2)$, with $T_2 >>> T_1$? In other words, is $\sigma$ always the right value to use regardless of when the bond matures?
2. I would like to obtain $\sigma$ from the market for pricing purposes, should I use historical volatility or the (risk-neutral) implied volatility? If implied, are there any well-known methods to obtain it for this scenario?

I sometimes find it a bit hard dealing with volatility when modeling. I have as a reference book "Volatility and Correlation" by Rebonato but I find it difficult to get direct answers from the book. If someone could point me to literature that covers what the best way of obtaining vol is for market practitioners, that would be very helpful.

## 1 Answer

The model has effectively two free parameters, and therefore one cannot expect it to match bonds of different maturities. Typically this is how you 'get the parameters' by solving to match a given set of instruments. Of course you cannot match 100 instruments with 5 parameters, therefore you either solve in a least squares sense or increase the number of parameters you calibrate. That depends on the purpose of fitting.

One might want to increase the degrees of freedom by making the drift time-varying, ie $\theta=\theta(t)$, which can then be calibrated to an array of bonds of different maturities.

Or make the volatility also time varying, ie $\sigma=\sigma(t)$, in order to also match some ATM swaption prices.

In practice this can be done through trinomial trees, google for "Hull White trees". The book by Brigo and Mercurio is IMO much better than Rebonato for recipes.

• Thanks for the reply, really helpful. However, let's assume that I would like to have a constant volatility instead of a time-varying one. Then the Hull-White tree would not be necessary, so how would I obtain the right $\sigma$ from the market? – Alfie Oct 15 '16 at 16:20