# How to calibrate Hull-White from zero curve?

I am interested in calibrating a Hull-White model to the market.

I do not, however, have data on anything except the market zero curves, as all derivatives are being traded OTC. My plan is to calibrate the model to the zero curve.

1. Will this produce a sensible calibration of the model in respect of derivatives?
2. If not, how does one proceed in this case?

The Hull-White model can represents the risk free rate as a stochastic process, that is, in terms of expected return and volatility. The zero curve only gives you expected returns and you have to find a source to calibrate volatility, as FQuant told you.

Common volatility sources used for this calibration are historical series of the zero curve or swaptions volatilities.

If you do not calibrate volatilities, you will not price correctly anything more complicated than a floating note.

Also please note two important short comings of the Hull-White model:

1. It allows negative rates, which implies an error on the pricing of non linear (or non floored) instruments.

2. If payoffs are defined in terms of several rates (or the same rate at different times) then your price is likely to be sensitive to correlations of these rates, and Hull-White is not able to represent these correlations.

Point 1. was not that an issue for 10 years but now the rates are pretty low, so that in Hull-White model driven simulations there is a large proportion of rates taking negative values in the first years of the simulation. (I once saw more than 30%!)

The one-factor Hull-White model is given by

$$dr(t) = (\theta(t) - \alpha\; r(t))\,dt + \sigma(t)\, dW(t)\,\!.$$

The zero curves are only sufficient for the calibration of the parameter $\theta(t)$, which is given in terms of them by

$$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2\alpha}(1-e^{-2\alpha t}),$$

where $f(0,T)$ is the instantaneous forward rate at time zero for a maturity of $T$. The parameters $\alpha$ (which describes mean reversion) and $\sigma$ (which describes volatility) have to be calibrated from external sources and chosen such that the output of the model, such as the price of zero-coupon bonds, match the market price of the instruments.

• The formulae you have written down define the price of a zero coupon bond. They are not related to the calibration of the model to the term structure of interest rates, which is achieved by the form of the parameter theta. Aug 13 '13 at 12:23
• I don't think this is correct. You can calibrate alpha and sigma via the explicit formulae for zero bonds, and then correct the arbitrage via theta? Aug 13 '13 at 12:37
• I am so sorry FQuant. I messed up my earlier comment and deleted it. So your comment seems out of place. Aug 13 '13 at 12:38
• So by zero curve, you mean the market data on zero bond prices? Aug 13 '13 at 23:53
• Yeah - i mean the zero coupon bond prices implied by the term structure... Aug 14 '13 at 10:12