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 =
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
- 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.