SABR Model

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This page covers the SABR model in depth. For context on how it fits into the vol surface pipeline, see How Surfaces Are Built. For a comparison with other methods, see Interpolation Methods.

SABR (Stochastic Alpha Beta Rho) is a stochastic volatility model introduced by Hagan, Kumar, Lesniewski, and Woodward (2002). Unlike SVI, which describes the shape of the smile, SABR describes the dynamics that produce it. The smile falls out of the model as a consequence of how vol evolves alongside the underlying.

SABR is the dominant model for interest rate swaptions and caps/floors. It is less common in crypto, where SVI is preferred for its simpler fitting and better wing behavior.

Explore the Parameters

Adjust each parameter to see how the SABR smile changes. Toggle "Show backbone" to see the CEV-only curve (what the smile looks like without vol-of-vol).

SABR Smile Explorer

Typical interest rate swaption smile. Moderate skew, mild curvature.

α (vol level)0.30

Current instantaneous vol

β (backbone)0.50

0 = normal, 0.5 = sqrt, 1 = lognormal

ρ (spot-vol corr)-0.30

Negative = put skew (usual)

ν (vol of vol)0.40

Controls smile width. 0 = no smile.

Toggle "Show backbone" to see the CEV-only curve (no vol-of-vol). The gap between backbone and full smile is the contribution of \u03BD.

What each parameter does

• $\alpha$ (vol level): The current instantaneous volatility. Higher $\alpha$ = higher overall IV. This is the most frequently recalibrated parameter.
• $\beta$ (backbone): Controls how volatility scales with the underlying price. $\beta = 1$ means percentage vol is constant (lognormal). $\beta = 0$ means dollar vol is constant (normal). $\beta = 0.5$ is in between (square root). In practice, $\beta$ is usually fixed to a market convention, not fitted.
• $\rho$ (spot-vol correlation): Controls the skew. Negative $\rho$ means vol rises when the underlying drops (the usual behavior in equities and crypto). Positive $\rho$ means the opposite (rare).
• $\nu$ (vol of vol): Controls the smile width. When $\nu = 0$, there is no smile, only whatever skew $\beta$ and $\rho$ produce (the "backbone"). As $\nu$ increases, both wings elevate.

The backbone

Click "Show backbone" in the explorer above. The dashed line is the smile with $\nu = 0$: no randomness in vol, just the deterministic CEV model. The gap between the backbone and the full smile is the contribution of $\nu$ (vol-of-vol). This decomposition is unique to SABR and gives traders a clear intuition for where the smile's curvature comes from.

Calibration

The standard approach

1. Fix $\beta$ to a market convention:

   • Interest rates: $\beta = 0.5$ (common) or $\beta = 0$ (normal SABR)
   • Equities: $\beta = 1$ (lognormal)
   • With $\beta$ fixed, the model has 3 free parameters.

2. Pin $\alpha$ to ATM vol. There is a near-closed-form relationship between $\alpha$ and ATM implied vol. Given the observed ATM IV, solve for $\alpha$. This reduces the fit to 2 free parameters.

3. Fit $\rho$ and $\nu$ by minimizing the weighted error between the SABR smile and observed IVs across strikes. With only 2 parameters, this is fast and robust.

Weighting

• ATM gets the most weight (most liquid, most reliable)
• Options with tight bid-ask spreads get more weight
• Deep OTM options get less weight (the Hagan approximation is less accurate there)

Strengths

Dynamic interpretation. SABR tells you how the smile should move when the underlying moves. By default, SABR produces a sticky-delta-like behavior: when spot drops, vol rises (if $\rho < 0$), and the smile shifts with spot. This is valuable for products where smile dynamics matter for hedging.

Backbone decomposition. The separation of backbone ($\beta$-driven skew) from smile ($\nu$-driven curvature) gives traders a clear mental model.

Parsimonious. With $\beta$ fixed and $\alpha$ pinned to ATM, you are fitting only 2 parameters. This is fast and leaves little room for overfitting.

Limitations

Wing problems. The Hagan approximation can produce negative implied vol or negative probability density in the far wings. This is a known issue. Production systems use corrected formulations (arbitrage-free SABR, or a PDE solver for extreme strikes).

Long maturities. The asymptotic expansion degrades for maturities beyond 10-15 years. Use a numerical method instead.

Static fit, not dynamic calibration. Despite SABR's dynamic interpretation, in practice each expiry is fitted independently (just like SVI). The dynamic story is aspirational rather than operationally enforced.

SABR vs. SVI

	SABR	SVI
What it models	Dynamics that produce the smile	Shape of the smile
Parameters	3 (with $\beta$ fixed)	5
Arbitrage	Hagan formula can violate in wings	Clean constraints available
Wing behavior	Can break for extreme strikes	Bounded, linear asymptotes
Speed	Formula evaluation	Optimization
Best for	Interest rates, FX	Equities, crypto

The key difference: SABR answers "how does the smile move?" while SVI answers "what does the smile look like?" For simple European option pricing and risk, SVI's simpler fitting and better wing behavior usually win. For products where smile dynamics matter (Bermudan swaptions, barrier options under sticky-delta), SABR's dynamic interpretation is valuable.

Connection to SVI

SABR can initialize SVI fits. Fit SABR first (fast 2-parameter optimization), evaluate the SABR smile at many strikes, then fit SVI to those points. This gives SVI a good starting point when market data is sparse.

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See also:

• SVI Parameterization - The model Hypercall uses
• Local Volatility - Dupire's instantaneous vol surface
• Interpolation Methods - All methods compared
• How Surfaces Are Built - The full pipeline