Calibrating Hull–White Volatility Beyond the Zero Curve

# How to calibrate Hull-White from zero curve?


# 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.

- Will this produce a sensible calibration of the model in respect of derivatives?

- If not, how does one proceed in this case?

## Answer by Michaël Le Barbier (score 11, accepted)

https://quant.stackexchange.com/a/9135

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:

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

- 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%!)

## Answer by FQuant (score 9)

https://quant.stackexchange.com/a/8734

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.