## Friday, May 25, 2012

### Volatility Smile in Rates | Understanding SABR Parameters

SABR is a standard skew model on the street for rates desk, but it's parameters and the part they play is often not very clear. Below is an approach for intuitive understanding of the parameters and how they affect pricing/calibration of collars and strangles

For strikes reasonably near the atm, we can have following approximation to good degree, for a normal bps vol and normal(horizontal when measured in bps vol) backbone

Skew = const1*correl*(volvol/atmvol)
Smile = const2*correl*(volvol/atmvol) + const3*(volvol/atmvol)^2

Here skew = (payer vol - rcvr vol)/atmvol, and smile = (payer vol + rcvr vol - 2*atmvol)/ atmvol

So the both the skew and smile depends on the relative strength of vol of vol to the ATM vol (or more precisely the local vol, but atmvol is a good proxy), i.e. on (volvol/atmvol). In case of skew this ratio acts through the correlation parameter. In case of smile, it is the square of the quantity that determines the wing vols. So as volvol/vol ratio goes down, the wings cheapen compared to ATM, and payers richen compared to receivers (given correlation is negative). And if correlation goes up, the skew steepen (on the payer side) given the ratio of volvol-to-vol is unchanged

Typically, this is a good approximation for reasonable strikes not too far away from the strikes

Also, for the smile, as the maturity goes up, the relative importance of the 2nd term (volvol to vol square) goes up compared to the 1st term, and so is the case as the width of the strangle goes up. So for wide long maturity stangle, (say 10y10y 200wide), the smile is practically determined by the 2nd part and correlation has practically zero impact. However, for short maturities they are of comparable importance and correlations and volvol gets entangled in both smile and skew. So a possible order of calibration can be first calibrating to the strangles followed by collars, recursively for long dated options and reverse for short dated options

To summarize the above, when looking at the sabr parameters, we should look at the correlations and volvol-to-vol ratio (relative strenght of volvol, NOT volvol itself)

In the recent rally that started March, we have seen a considerable richening of the payer skew. Upper-right, coming off from super-normal, upper left still sticking to log-normal

In the upper-right, in spite of a marked change in the realized correlation downwards, skew has moved up. At the same time, the volvol to vol ratio came down with increasing vol. So in effect, skew richened and the smile flattened out

With the Greece event risk in the near horizon, this cheapening of smile and receiver side skew represents an opportunity for play on the receiver sides with spreads/flies

I have left out another parameter of the SABR model from here, i.e. BETA. I will have another post to discuss that in details