SABR Model

The SABR model, introduced by Patrick Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward in 2002, is the standard way to model volatility smiles in interest rate options. Its name stands for "stochastic alpha, beta, rho", three of its parameters. SABR describes how a forward rate or price and its volatility move together, and it comes with a simple approximate formula for implied volatility at each strike. That formula made it practical for traders to fit and interpolate smiles for swaptions, caps and floors, and it is widely used in FX and commodity markets too.

The model#

SABR models a forward F and its volatility α:

dF = α × F^β dW1
dα = ν × α dW2
corr(dW1, dW2) = ρ

The forward has no drift because it is a forward under its own pricing measure. Volatility follows a lognormal process with no mean reversion.

The four parameters#

β deserves special attention:

• β = 1: lognormal behaviour, like Black 76.
• β = 0: normal behaviour, like Bachelier, allowing negative rates.
• β = 0.5: a common compromise in rates markets.

See Black-76 and Bachelier Models.

Hagan's formula#

The key practical result is an approximate closed form expression for Black (or normal) implied volatility at any strike, given the four parameters. Traders calibrate α, ρ and ν (with β fixed) to market volatilities for each expiry and tenor, then use the formula to read implied volatility at any strike.

Why SABR caught on#

• Realistic smile dynamics: when the forward moves, the smile moves with it in a way that matches market behaviour better than local volatility, which improves hedging. This was the main motivation of Hagan's paper, "Managing Smile Risk". See Local Volatility.
• Simple, fast formula for implied volatility.
• Few, intuitive parameters that traders can relate to level, skew and curvature.
• Flexibility across rates, FX and commodities.

Known problems and fixes#

• Negative densities at low strikes: Hagan's approximation can imply negative probabilities for very low strikes, especially for long expiries, which creates arbitrage. Fixes include arbitrage free SABR methods solving the model numerically.
• Negative rates: the original model with β > 0 cannot handle negative forwards. Shifted SABR adds a constant to rates; normal SABR uses β = 0.
• No mean reversion: volatility can drift without bound over long horizons, which matters for long dated products.
• One expiry at a time: SABR is usually calibrated separately for each expiry, so it is a smile model rather than a full term structure model.

SABR vs Heston#

See Stochastic Volatility and the Heston Model.

Frequently asked questions#

What is the SABR model?#

A stochastic volatility model for forward prices or rates, with four parameters that control the level, backbone, skew and curvature of the volatility smile.

What does beta mean in SABR?#

It sets how volatility relates to the level of the forward: β = 1 is lognormal, β = 0 is normal, and values in between blend the two.

Why is SABR used in interest rate markets?#

Because it fits swaption and cap smiles well with a fast formula and gives realistic smile movements for hedging, with shifted or normal versions handling negative rates.

Next, learn how traders fill gaps between quoted options in Volatility Interpolation and Extrapolation.

Sources#

• Wikipedia, SABR volatility model