SABR Skew Dynamics and Beta | Part - I

Dealers in swaptions usually witness a particular skew dynamics - as the market sells off, the receiver skew richens and similarly a rally usually accompany a richening of the payer skew. So a plot of change in rcvr skew vs change in atmfs should look like an upward sloping line. The real life reason for this dynamics can tentatively be either from mean-reverting trade demands (buying payers on the rally etc) or due to sticky strike vols (large positions pinned around standard strikes).

The market standard model, SABR, successfully models this particular skew dynamics. This modeled change in skew in absence of change in other parameters is due to two parts - 1) the dvol/drate and 2)the SABR beta parameters. However, recently there is much stronger relationship between skew changes and ATMF (at the money forward rate) changes in the modeled dynamics than what is actually realized - especially for longer tenors and longer expiries - more so if we consider these are the points where dvol/drate is quite insignificant. 

This reflects a stronger-than-actual skew/rates dynamics modeled by SABR. A large part of SABR modeled skew dynamics is reversed on re-calibration to bring it back to the realized levels. This is especially true for longer expiries and longer tenor options.

SABR is an over-parameterized model and it is not possible to calibrate all the four underlying parameters from a snap-short of market prices. The usual practice is fix the beta parameter and calibrate the others. Which is not really an issue, esp if the model is treated as more like a good interpolater of vols, rather than a good model of the underlying dynamics itself, and remarked continuously. However, remarking continuously is usually not possible.

For delta-hedged market-makers this raises the issue of beta-induced market dynamics. The prediction of change in skew for a given change in rate, with other parameters remaining the same, is driven mostly by beta as mentioned above and the relationship is as follows 

change_in_skew = const*atmvol*(1-beta)*change_in_atmf, 

For a given expiry, change in skew increases as the time to expiry goes down. As a results closer the backbone parameter towards log-normal, greater the change in modeled skew (in terms of normal bps vol, which is market standard in rates space). This also mean a marked blend lower than the "actual" blend will overestimate the skew changes and vice versa. Currently, the 10y tails skew dynamics is over-estimated by SABR. At the current marks, the 5y tails shows good agreement with recent market dynamics. This has implications for delta-hedging, as SABR will miscalculate the delta profiles of long tenor trades, especially ones with high dVega/dRate. And this will mean surprises for the dealers, especially following large market moves (and little change in vols) - the kind of markets we are witnessing recently