# zero-shift SABR vega and re-calibration of SABR

I have a zero-shifted SABR model, where I need to confirm if the model is generating the calibration and vega's correctly.

The underlying model is the standard SABR lognormal (there is normal as well), with the forward = Forward - zero-shift.

Given an implied volatility curve (2yr expiry, 5yr tenor for example), where I have calibrated the z-shift SABR parameters of ATMvol, rho-vol, vol_of_vol, beta (alpha, rho, volvol, beta)

1. if I increase the z-shift from 100bp to 200bp (i.e. Forward = Forward- 200bp)
2. Given an existing long swaption position (2y expiry, 5yr tenor), strike of 2.50% which is 150bp away from ATMF of 1.00% as of today.

How might a fixed-strike Swaption vega's and calibrated parameters change?

My intuition is as follows

1. For vega's : the rho (correlation) vega increases, the vol-of-vol vega increases, the ATM vega does not increase. This is mainly because at the fixed 2.50% implied volatility, when we increase the z-shift, we take the effective ATMF volatility point further away from the 2.50% implied vol-point. Hence, a similar change (increase) in rho and vol-vol will increase the implied volatility at 2.50% much more.

2. For re-calibration, I am not sure if the same logic applies.

Any expert thoughts out there?

Kind regards

performed an experiment myself, using Bachelier's Black Model and coded shifted-SABR normal model. I observe the following

We perform a series of experiments, that tests for

•  Given different z-shifts, what are the SABR parameters to calibrate to the target-set of implied vols
•  What parameters are necessary and how do they change.

With that, we propose a pseudo set of target vols (similar in nature to Inflation-level year-on-year vols HICP). With the Fwd at 2.97%, Time of 1yr tenor, and initial z-shift of 1.00%. We define the z-Forward = Forward (2.97%) + z-shift (1.00%). We initially set the free parameters as only rho and vol-vol; with beta at 0.01 (we prefer not to set to 0.0 explicitly). We generate for z-shifts from 1.00% (initial) to 3.00%. See table1 and chart 1 below. We observe that

•  These two parameters are not enough for an absolutely good fit (error increases), though deemed acceptable.
•  Rho becomes less and less negative (i.e it increases) while vol-vol becomes less and less positive (also smaller in magnitude). We infer that (given higher-strike is lower-in-vols), the negative skew becomes less and less needed as we shift the ATMF towards the (absolute strike point). This is because a smaller rho (and vol-vol) are required to hit the extreme higher-strike vols. We might argue, the opposite should occur for the lower strike vols. We now set the free parameter to be alpha (ATM vols), rho and vol-vol. We observe the fit is now very good, with the load of hitting being carried by alpha. See table and charts 2.

•  We observe the vol-of-vol does not change significantly while
•  The ATM vols and rho changes significantly.

We observe and try to explain the changes in SABR parameter vega’s. We take two cases,

•  Base-case of z-shift = 1.00%, with SABR calibrated to the target volatilities. We then generate the SABR vega’s, of call-options from strikes 0.00% to 7.00% with ATMF = 3.00%
•  New-case of z-shift = 2.00%, with SABR re-calibrated to the target vols (alpha, rho and vol-vol, but not beta). We re-generate the SABR vega’s using the same 0.00% to 7.00% strike swaptions with ATMF = 3.00%

Both we apply the Bachelier Normal Black formula, with notional of 1,000,000.

We observe and propose they make sense on the basis

•  Alpha : equivalent to ATM vols, and hence replicate the standard vega profile.
•  Rho : for swaptions/options above ATMF, they are long rho, and short rho for options below ATMF (3%).
•  Volvol : should be positive across all strikes, as all implied vols increase.
•  Beta : shifts from normal to lognormal with an effective increase in rho and hence roughly similar in risks to rho.