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Advanced Derivatives and Exotic Options / SABR and Volatility Asymptotics

The SABR Model — Stochastic Alpha, Beta, Rho, and the Hagan Approximation

SABR (Hagan, Kumar, Lesniewski, Woodward, 2002) is the standard model for interest rate derivatives smiles. Unlike the Heston model — where variance follows a mean-reverting CIR process — SABR models the forward FtF_t with CEV dynamics and lets the volatility itself follow a lognormal process. The result: a four-parameter model with a closed-form approximation for the implied vol smile that rates desks use in production for swaptions, caps/floors, and CMS rates.

The Hagan approximation is an asymptotic expansion in total variance α2T\alpha^2 T — accurate when options are not too long-dated and not too far from the money. Its simplicity (a single formula for the full smile at each expiry) made SABR the rates standard. For equity, SABR is less common — Heston better captures the strong mean-reversion and forward-skew properties of equity markets — but the SABR framework's asymptotic insight (that vol dynamics generate specific smile shapes) extends throughout the unit.

Learning objectives. By the end of this note you should be able to:
  • write the SABR SDE system and identify the role of each parameter;
  • state the Hagan approximation for implied vol and compute it numerically;
  • calibrate SABR to a single maturity's smile (ATM vol fixes α\alpha; skew and curvature fix ρ\rho and ν\nu; β\beta is often pre-specified);
  • explain why β\beta is typically fixed rather than calibrated.

In this lesson

  1. 4.1 The SABR model
  2. 4.2 The Hagan approximation
  3. 4.3 Calibration procedure

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