Re: SABR model

Posted by spz on Nov 26, 2010; 4:09am
URL: http://quantlib.414.s1.nabble.com/SABR-model-tp265p266.html

Hi Eduardo,

I think it was a tiny bug in the implementation. It should be a Taylor expansion about z = 0 (when close to ATM, z is close to 0 ). I did the expansion, and found the following formula

z/xx ~ 1 - 0.5*rho*z + (3*rho*rho + 6*rho + 2)/12*z*z

The difference of this one and the one in the implementation is 0.5*rho*rho/(1-rho)*z in leading order. Actually, when z is small ( z*z < QL_EPSILON*10 ), this difference doesn't matter (maybe a relative error of 1e-6 at most). This is why I call it a tiny bug.

I did check the formulas visually using Matlab, both expansion gives extremely close result. The new formula I showed is better than the original QL implementation only when z is greater than 1e-5, but that's not the job of the Taylor expansion formula as z is big enough.

Peizhe

Eduardo Alonso wrote

Dear all, I have a problem with the sabr model ( implementation in
ql/termstructures/volatility/sabr.cpp).
There is an approximation for close to at the money that I still don't know
where it comes from. The multiplier is z/xx, but close to ATM there is what
looks like a Taylor expansion but I couldn't figure out what is being
expanded and about what point. Can someone tell me what's going on there? I
am using Hagan's paper on SABR and I see nothing similar. If you can give me
another reference where that approximation is to be found, or if you can
tell me how to deduce it, I would appreciate.

Thanks a lot in advance and regards to everyone.

Eduardo

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