SABR (Stochastic Alpha Beta Rho): joint diffusion of forward and vol
Hagan 2002: jointly stochastic forward and vol-of-forward, with one closed-form formula for implied vol covering the whole smile. Standard equity, rates, and FX skew calibration.
A rates or FX desk prices hundreds of options per day across different strikes — all quoting in Black-implied vol terms. To price consistently across the whole smile, the desk needs a model that outputs an implied vol for any strike in microseconds. SABR (Hagan-Kumar-Lesniewski-Woodward 2002) delivers exactly that: a two-SDE model for the forward and its volatility, with a closed-form approximation for the Black-equivalent vol at any strike. No simulation, no numerical integration. The four parameters $(\alpha, \beta, \rho, \nu)$ each control a distinct feature of the smile: $\alpha$ is the ATM vol level, $\beta$ sets the backbone shape (0 = normal, 1 = lognormal), $\rho$ tilts the skew, and $\nu$ is the vol-of-vol that fattens the wings. Calibrate once per maturity slice and you get a near-instantaneous read-out for any strike.
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Worked example
An equity-style SABR fit has $\alpha = 0.80$, $\beta = 0.7$, $\rho = -0.3$, $\nu = 0.4$, with forward $F = 100$ and maturity $T = 1$. Use the Hagan 2002 closed-form formula (oracle) to compute the at-the-money implied vol $\sigma_{BS}(K = 100)$.
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Reflection
SABR's $\beta$ parameter is notoriously under-identified by smile data alone: many $(\alpha, \beta)$ pairs produce visually indistinguishable smiles. Why is this a problem in practice, and what's the standard workaround?