Why is SABR the standard for swaption volatility?

Question

AcadiFi · Accepted Answer

SABR — Stochastic Alpha Beta Rho — is a parametric stochastic volatility model designed for parsimonious smile fitting at a single maturity. Its four parameters (alpha, beta, rho, nu) are intuitive and stable across calibrations, which is why swaption desks love it.

The dynamics: dF = alpha * F^beta * dW1, d(alpha) = nu * alpha * dW2, with correlation rho. For each swaption expiry-tenor pair (say 5y-into-10y), a separate SABR surface is fit.

Parameter roles:
- alpha: ATM vol level
- beta: CEV exponent (0=normal, 1=lognormal, often fixed at 0.5)
- rho: skew control (negative rho tilts smile downward)
- nu: vol-of-vol, controls smile convexity

Luka Trubetskaya at Warsaw Fixed Income Quants calibrates SABR on 1,920 swaption grid points daily. The Hagan approximation formula gives closed-form Black implied vols from SABR parameters, enabling instant repricing and hedging.

```mermaid
flowchart TD
  A[Market Swaption Vols] --> B[Fit SABR per Expiry-Tenor]
  B --> C[alpha: ATM]
  B --> D[beta: CEV]
  B --> E[rho: Skew]
  B --> F[nu: Vol-of-Vol]
  C --> G[Smile Surface]
  D --> G
  E --> G
  F --> G
```

SABR's weaknesses: negative rates break beta=1 (lognormal), so shifted SABR or normal SABR (beta=0) is used. At low strikes, Hagan's formula can produce arbitrage (negative densities). Modern variants like Antonov-Konikov-Spector correct this. Despite flaws, SABR remains dominant because it's interpretable, stable, and closed-form — properties that matter for a desk marking thousands of swaptions at 4:30 PM London fix.

Why is SABR the standard for swaption volatility?

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