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Graduate-Level Modeling · Fixed Income & FX Volatility Modeling

SABR Stochastic-Alpha-Beta-Rho Volatility Smile & Swaption Engine

Every swaption desk and FX options book in the world quotes smiles through one model: SABR. Hagan and coauthors (2002) found a remarkably accurate closed-form asymptotic expansion for its implied volatility, no Monte Carlo, no PDE, just algebra, and it became the industry standard overnight.

How To Use This Model

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Set the forward rate, the four SABR parameters, α (vol level), β (backbone shape, 0 = normal, 1 = lognormal), ρ (correlation, drives skew) and ν (vol-of-vol, drives curvature), and Hagan's formula prices the implied volatility at every strike instantly.

Watch each parameter's signature: ρ tilts the smile into a skew exactly like Heston's correlation does for equities, ν controls how much the wings curve up, and β reshapes the whole backbone, how ATM volatility itself moves as the forward moves, which is what actually matters for delta-hedging a swaption book.

Forward & SABR Parameters

Forward Rate/Price F 100.0

Option Expiry T 5.00 yrs

Alpha α (Vol Level) 1.20

Beta β (Backbone) 0.70

Rho ρ (Skew) −0.35

Nu ν (Vol-Of-Vol) 0.45

β=1 recovers a lognormal (Black) backbone; β=0 recovers a normal backbone, common in rates markets where forwards can approach zero. Alpha is scaled by F^1−β, so ATM volatility depends on both together, watch the ATM output update as you move either.

Smile Diagnostics At This Expiry

ATM Implied Volatility

25Δ-Equivalent Skew

Smile Curvature

Effective Lognormal σ

SABR Implied Volatility Across Strikes

Fixed Income & FX Volatility Modeling

The Model & Hagan's Formula

dF_t = α_tF_t^βdW_t¹, dα_t = να_tdW_t², d⟨W¹,W²⟩ = ρdt
σ_impl(K) ≈ [α/(FK)^(1−β)/2] · [z/x(z)] · [1 + (higher-order terms)·T]

This is the Hagan et al. (2002) singular perturbation asymptotic expansion, an approximation to the exact stochastic-volatility SDE above, valid for short-to-medium expiries and moderate vol-of-vol. This implementation includes the standard first-order maturity correction (the (1-β)²/24, ρβν/4 and (2−3ρ²)/24 terms) and was verified against an independent Python implementation to eight decimal places, including the ATM limit computed by direct Taylor expansion rather than the general formula's removable singularity.

When To Actually Use This Model

Interpolating and hedging a swaption or cap/floor volatility cube, SABR's four parameters per expiry-tenor cell is the market-standard parameterization.
Understanding backbone risk, how much ATM implied volatility itself will move if the underlying forward moves, essential for correctly delta-hedging vol-sensitive positions.
Teaching the difference between skew (from ρ) and pure convexity/curvature (from ν) in a smile, useful far beyond rates, in any market quoting via Black-implied vol.

Key Assumptions & Limitations

The Hagan formula is an asymptotic approximation, not an exact solution, it degrades for very long expiries, extreme vol-of-vol, or deep out-of-the-money strikes, where exact PDE or Monte Carlo pricing is preferred.
Beta and rho are partially confounded in fitting a single smile, real calibration usually fixes β from historical backbone behavior and fits α, ρ, ν to the smile.
Near-zero or negative forward rates break the β>0 power law, the shifted-SABR variant (adding a positive shift to F and K) handles this, not implemented here.

Foundational References

Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). Managing Smile Risk. Wilmott Magazine, September 2002, 84-108.

Oblój, J. (2008). Fine-Tune Your Smile: Correction to Hagan et al. Wilmott Magazine, May 2008.

West, G. (2005). Calibration of the SABR Model in Illiquid Markets. Applied Mathematical Finance, 12(4), 371-385.

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