Why is SABR the standard smile model on a swaption desk, what does the beta parameter control, and what went wrong with the standard formulation when rates went negative?

Why is SABR the standard smile model on a swaption desk, what does the beta parameter control, and what went wrong with the standard formulation when rates went negative?

Approach: Start from what a swaption grid needs, a smile per expiry and tenor that can be recalibrated fast, then look at what the backbone assumption does when the forward approaches zero.

SABR is standard because it gives a closed-form implied volatility for each expiry and tenor with four intuitive parameters, and beta sets the backbone, meaning how the at-the-money volatility moves as the forward moves. Beta of 1 makes the process lognormal so at-the-money volatility is roughly unchanged as the forward shifts, beta of 0 makes it normal so the lognormal volatility moves inversely with the forward, and desks usually fix beta by convention and fit alpha, rho and the volatility of volatility to the market. The formulation broke when forwards went negative because a lognormal or a square-root process cannot produce a negative rate at all, so the model had no price to quote for a large part of the grid. The two fixes in use are the shifted lognormal SABR, where the forward plus a shift of 2% or 3% is modelled, and quoting in normal volatility outright, which handles zero and negative forwards without a shift. Normal volatility became the market convention in several currencies for exactly that reason.

Follow-up: The asymptotic SABR expansion misprices very low strikes and can imply a negative density. How would you detect that and what would you do about it?

Key concepts: SABR model, backbone, beta parameter, normal volatility, shifted lognormal.