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### Hull-White and skew

Posted: **February 4th, 2011, 3:39 pm**

by **donal**

Hi all,I'm just beginning to look at interest rate modelling, starting with Hull-White. I'm somewhat confused and wondered if anyone could shed any light on where my thinking is going wrong.As I understand it, you can calibrate the model to ATM swaption vols e.g. from VCUB on Bloomberg. Now, if I want to price a swaption with a strike away from the ATM forward rate, my thought is that I must need to factor in some sort of IR skew, but I don't understand how the HW model does this? I guess my thinking follows from how we would look at equity options i.e. I wouldn't price a 110% call on an equity with the ATM vol.Any thoughts would be appreciated.Cheers,Donal

### Hull-White and skew

Posted: **February 4th, 2011, 4:27 pm**

by **mynetself**

Kinda interested in the (general) topic myself. As I understand it, once you've calibrated your model to a certain set of financial instruments, that's it. If you then use it to price stuff, you should price the ones you used for calibration exactly, but everything else is a "prediction" of the model. Not much you can do about it.

### Hull-White and skew

Posted: **February 4th, 2011, 5:31 pm**

by **Costeanu**

Hi donal, Hull-White cannot match the skew. Generally for short-tenor swaptions the smile implied by Hull-White is closed to normal, and it tilts a bit for longer tenor, but you cannot manipulate it by playing with the model parameters. Well, technically you could for long-tenor swaptions, by playing with the speed of mean reversion, but you shouldn't attempt to do that at home This is a price you have to pay for having closed form formulas for zero-coupon bonds. There are two ways out of this: either use out-of-model adjusters (search for Hagan adjusters) or use a model that has smile. The two models that have smile and also closed-form zero-coupon bonds are the quasi-Gaussian model (a.k.a. Cheyette) and the quadratic Gaussian model. The first is a straightforward generalization of HW, but you pay a price in the number of state variables. The second is again a generalization of HW, where you add a quadratic term, as the name implies. However, it requires some thougth to get familiar with. Of course, there are models without closed-form ZCB; you calculate the ZCB's on the grid using backward induction. Best,V.

### Hull-White and skew

Posted: **February 4th, 2011, 6:12 pm**

by **Cuchulainn**

QuoteOf course, there are models without closed-form ZCB; you calculate the ZCB's on the grid using backward induction. Using a binomial lattice?

### Hull-White and skew

Posted: **February 4th, 2011, 7:04 pm**

by **Costeanu**

I suppose a binomial tree works. In general people use trinomial trees, or more advanced trees (willow and SALI). You can probably use Crank-Nicolson as well, or some other finite difference scheme.