Re: [Quantlib-users] SABR model

[spz <shi...@gm...>]

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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