Friday, September 13, 2013

SABR Skew Dynamics & Beta | Part II

In original SABR model, the beta has two distinct roles - 1) it determines the back-bone of the atmvol vs atm and 2) it contributes to beta skew (and beta smile) in a way that is independent of the vanna smile (the smile driven by correlation parameter).

However, in most common implementations of SABR, this is NOT the case. The traders in rates are familiar in normal bps vols. So the ATM vols are input in terms of normal bps vols (as opposed to the ALPHA parameter, the ATM vol in correct beta).

Because of the implementation, the beta COMPLETELY looses its first role (the modeled back-bone). Instead of determining the the dynamics in terms of the back-bone, beta switches to play its role in determining the skew dynamics as mentioned in below mail - which will not be seen in a true implementation 

On the 2nd role as well, the implementation introduces a twist. The main point in implementation is, as opposed to parameterizing the model in terms of ALPHA (local vol in the correct blend), CORREL. VOLVOL and BETA, it parameterizes in terms of ATMVOL (vol in normal blend), CORREL, VOLVOL and BETA. in a generic model the interplay between beta skew and correl skew depends on two parameters - BETA and CORREL*LAMBDA - where LAMBDA is ratio of VOLVOL/ALPHA. So while the general rule is that a instead of decreasing the correlation (more negative) one can increase the beta to match a skew, this has impact on far OTM strikes.

As one increases the beta, the local-vol (at the correct blend) falls drastically, This drastically changes the LAMBDA ratio. This introduces a very different smile (curvature) effect at far OTM strikes This shows 1y10y smile for 1) correl =0, beta = 0.35 (blue) vs 2) correl=0.55, beta =1 (green). As can be seen the curvature on the far OTM sides is very different. We match the skew in 2nd case by setting beta =1 and increasing the correl. However this changes the local vol and the LAMBDA ratio. This introdcues difference in smiles in the far OTM strikes Typically, matching skews by changing beta or correl is NOT equivalent. One difference is obviously the change in the dynamics (which will impact hedges) mentioned in the previous post on this. The other difference, as illustrated above, a change in OTM smile

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