The SABR Model

SABR, Stochastic Alpha Beta Rho, is the model rates and FX desks actually use to quote and risk-manage the smile. Its selling point is not a philosophically superior model of returns but a practical one: Hagan's 2002 asymptotic formula gives the entire implied-vol smile in closed form as a function of four intuitive parameters. That turns smile-fitting into a fast, stable interpolation and, crucially, gives smile-consistent Greeks, which is why it displaced ad-hoc interpolation on every swaption and cap/floor desk.

The SABR SDEs

SABR models a single forward $F_t$ (a forward rate, forward price, or swap rate) with a CEV local elasticity and a lognormal stochastic volatility:

$$\begin{aligned} dF_t &= \alpha_t\,F_t^{\beta}\,dW_t^{1},\\ d\alpha_t &= \nu\,\alpha_t\,dW_t^{2},\\ dW_t^{1}\,dW_t^{2} &= \rho\,dt. \end{aligned}$$

Note it is a forward-measure model, so the forward is driftless (a martingale). The four parameters:

• $\alpha$, the initial volatility level (overall vol; sets ATM vol).
• $\beta \in [0,1]$, the CEV/backbone exponent: it fixes how ATM vol moves as the forward moves. $\beta = 1$ is lognormal (vol level-independent), $\beta = 0$ is normal/Gaussian (used for low or negative rates), $\beta = \tfrac12$ is the CIR-like square-root case common in rates. $\beta$ is usually fixed a priori (by market convention or historical backbone), not fitted.
• $\nu$, the vol-of-vol: controls the smile's curvature (convexity), exactly like $\xi$ in Heston.
• $\rho$, the correlation between forward and vol: controls the smile's skew/slope.

So $\beta$ and $\rho$ together set the slope; $\nu$ sets the curvature; $\alpha$ sets the level. This clean separation is why traders find SABR intuitive.

Hagan's implied-volatility expansion

SABR's fame is the singular-perturbation (small-time) expansion Hagan et al. derived for the Black (lognormal) implied vol $\sigma_B(K,F)$. For strike $K$ and forward $F$:

$\sigma_B(K,F) = \frac{\alpha}{(FK)^{(1-\beta)/2}\Big[1 + \tfrac{(1-\beta)^2}{24}\ln^2\tfrac{F}{K} + \tfrac{(1-\beta)^4}{1920}\ln^4\tfrac{F}{K}\Big]}\cdot\frac{z}{x(z)}\cdot\big[1 + (\dots)\,T\big],$

where the smile-shaping factor uses

$z = \frac{\nu}{\alpha}\,(FK)^{(1-\beta)/2}\ln\frac{F}{K}, \qquad x(z) = \ln\!\left(\frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho}\right),$

and the $O(T)$ bracket is

$1 + \left[\frac{(1-\beta)^2}{24}\frac{\alpha^2}{(FK)^{1-\beta}} + \frac14\frac{\rho\beta\nu\alpha}{(FK)^{(1-\beta)/2}} + \frac{2 - 3\rho^2}{24}\nu^2\right]T.$

As $K\to F$ (at-the-money) this simplifies to the much-used ATM SABR vol:

$\sigma_{\text{ATM}} = \frac{\alpha}{F^{1-\beta}}\left[1 + \left(\frac{(1-\beta)^2}{24}\frac{\alpha^2}{F^{2-2\beta}} + \frac14\frac{\rho\beta\nu\alpha}{F^{1-\beta}} + \frac{2-3\rho^2}{24}\nu^2\right)T\right].$

The factor $z/x(z)\to 1$ as $z\to 0$, so near the money the smile is governed by the leading term and the skew enters through $\rho$ and $\beta$. The formula is explicit, no ODEs, no integrals, so an entire smile evaluates in microseconds, and its derivatives give smile-aware Greeks.

Why rates and FX desks use it

• Closed-form smile. Every swaption, cap, and floor smile is quoted and interpolated with one Hagan formula; no numerical PDE per quote.
• Intuitive, stable parameters. $\alpha$ (level), $\rho$ (skew), $\nu$ (curvature) map directly to what traders see and hedge; $\beta$ is a modelling choice tied to the observed backbone.
• Smile-consistent risk. SABR gives a dynamic smile: as the forward moves, the model tells you how ATM vol and the whole smile shift, so the SABR delta and vega are corrected for smile movement (the famous vanna-adjusted "SABR delta"). This backbone behaviour is the real reason it beat sticky-strike interpolation.
• Negative rates. The shifted SABR ($dF = \alpha(F+s)^\beta dW$) or $\beta = 0$ normal SABR handles the negative-rate world that lognormal models cannot.

Worked example

A 1-year swaption: forward swap rate $F = 3\%$, and the desk fixes $\beta = 0.5$. Calibrate to three quotes, ATM vol and two wings, recovering, say, $\alpha = 0.015$, $\rho = -0.3$, $\nu = 0.4$. The ATM lognormal vol is approximately $\sigma_{\text{ATM}}\approx \alpha/F^{1-\beta} = 0.015/0.03^{0.5} = 0.015/0.173 = 8.7\%$ (before the small $O(T)$ correction). The negative $\rho = -0.3$ tilts the smile so low strikes (receiver swaptions) carry higher vol; $\nu = 0.4$ sets the wing curvature. Bumping $F$ up moves the whole smile per the $F^{1-\beta}$ backbone, giving the SABR delta.

What breaks in practice

• Arbitrage in the wings at low strikes. Hagan's expansion is only asymptotic; for long maturities or strikes near/through zero it produces negative densities (a butterfly-arbitrage). The Hagan "arbitrage-free SABR" (Hagan 2014, solving the density PDE) or Antonov's exact/mapping formulas fix it.
• Small-time expansion. The formula degrades for long-dated options where the $O(T)$ term is no longer small; higher-order or exact methods are needed.
• $\beta$ is nearly unidentifiable from the smile alone. $\beta$ and $\rho$ both affect skew and trade off against each other, so $\beta$ is set by convention/backbone, not fitted, fitting all four is ill-posed.
• Single forward. Basic SABR prices one expiry-tenor smile; a full term structure needs SABR-LMM or a stochastic-vol LMM, and joint calibration is hard.

In interviews

State the SABR SDEs and the role of each parameter: $\alpha$ = level, $\beta$ = backbone/CEV (fixed, controls how ATM vol moves with the forward), $\rho$ = skew, $\nu$ = vol-of-vol/curvature. The headline is Hagan's closed-form implied-vol formula, you don't need to memorize every term, but know it exists, that it makes smile-fitting an instant interpolation, and that $z/x(z)$ carries the skew. Explain why rates/FX desks prefer it: intuitive parameters, closed-form smile, smile-consistent (backbone-aware) Greeks, and shifted-SABR for negative rates. The key caveat: the asymptotic formula can generate arbitrage in the low-strike wing, fixed by arbitrage-free SABR. Contrast with Heston (Fourier, equities) and local vol (exact vanilla fit, wrong dynamics).