What is Shifted lognormal model, shifted SABR?

Example 5.5 (One smile, three models)

A one-year option on a two-year euro swap, forward $-0.30\%$, is quoted at normal volatilities of 29.3, 26.5, 25.7, 27.8, 31.4 and 39.8 basis points at strikes 50 and 25 basis points below the forward, at the money, and 25, 50 and 100 above (a smile generated by normal SABR, illustrative). Shifted SABR with $\beta=0.5$ fits it to 0.77 basis points of root-mean-square error with a shift of 1%, to 0.11 with a shift of 3%; normal SABR fits it exactly. Asked for a receiver struck at $-0.90\%$, sixty basis points below the forward and outside the quotes, they answer 0.177, 0.280 and 0.295 basis points of annuity (Figure 5.1): with a 1% shift the modelled rate cannot fall below $-1\%$, and the deep receiver is 37% cheaper than with 3%.

Example 5.8 (A synthetic euro cube)

Sixteen sections (expiries one, two, five and ten years; tenors two, five, ten and thirty years), seven strikes each from $-200$ to $+200$ basis points around the forward on chapter 2’s curves, generated by normal SABR with parameters that vary smoothly with expiry and tenor plus seeded noise of 0.3 basis point, are calibrated with shifted SABR ($\beta=0.5$, shift 3%). Every section fits to between 0.21 and 0.78 basis point. The at-the-money face (Figure 5.3) rises with expiry and falls with tenor: 69.9 basis points for one year into two, 74.0 for ten into thirty.

Example 5.11 (Ten years into ten)

A ten-year option on a ten-year swap at 2.50%, quoted from $-200$ to $+200$ basis points around the forward (86 to 96 basis points of normal volatility), is fitted by shifted SABR with a shift of 1% (error 1.28 basis points) and of 3% (0.35). With the 1% shift the density is negative from $-0.95\%$ to $+0.35\%$, reaching $-158$; with 3% it stays positive down to $-2.15\%$ (Figure 5.4). A book of low-strike receivers priced with the 1% fit carries an arbitrage against the market’s butterflies.