SABR

The smile model of the rates and currency desks. Four parameters per maturity, a closed-form implied volatility from Hagan's expansion, and a fit that takes milliseconds. Set the parameters and read the smile, or fit SABR to a market slice and see how much of the shape four numbers explain.

Parameters

At-the-money volatility

α is solved from this

Backbone β

Correlation ρ

Tilts the smile

Vol of vol ν

Bends the smile

Forward 105.13 from spot 100.00, rate 5.0%, yield 0.0%, maturity 1 years. α = 0.1966.

Market

Spot (S)

Maturity (T)

Rate (r)

Yield (q)

Fit to a market slice

Market shape

Try this

0 of 3 done

Level, tilt, bend

Set ν to 0 and read the smile, then ρ to -0.5 with ν = 0.5, then ρ to 0 with ν = 1.

What to look for: With ν at zero the smile is flat at the at-the-money volatility. ρ tilts it; ν lifts both wings. Each parameter owns one feature of the shape.

Fit a market

Choose the equity-skew market and fit.

What to look for: The error lands well under a vol point and the fitted ρ is negative. Four numbers reproduce the whole slice, which is why the fit can be redone on every tick.

β is a convention

Fit with β = 1, note the error, then set β = 0.5 and fit again.

What to look for: The error barely changes while ρ and ν shift to compensate. The data cannot tell β apart from ρ, so desks fix it and fit the rest.

β sets how the at-the-money level moves when the forward moves, the backbone; ρ tilts the smile and ν bends it. α is solved so the at-the-money volatility stays where you set it.