A model per maturity

Heston and Bates describe the whole surface with one set of parameters. Rates and currency desks want something lighter: a smile for each maturity that can be fitted in an instant and re-fitted every time a quote changes. Hagan, Kumar, Lesniewski and Woodward (2002) gave them SABR, "stochastic alpha, beta, rho":

$$ dF_t = \alpha_t F_t^{\beta}\, dW^{(1)}_t, \qquad d\alpha_t = \nu\, \alpha_t\, dW^{(2)}_t, \qquad dW^{(1)}_t\, dW^{(2)}_t = \rho\, dt $$

• \(\alpha\): the volatility level, solved so the at-the-money quote is matched.
• \(\beta\): the backbone. With \(\beta = 1\) the forward is lognormal; with \(\beta = 0\) it is normal and can go negative, which rates markets need. It is fixed by convention, not fitted.
• \(\rho\): the correlation between the forward and its volatility, which tilts the smile.
• \(\nu\): the volatility of volatility, which bends it.

The formula

The model's gift is an asymptotic expansion for the implied lognormal volatility at any strike, accurate for the maturities and volatilities where it is used:

$$ \sigma(K) \approx \frac{\alpha}{(FK)^{(1-\beta)/2}\left[1 + \frac{(1-\beta)^2}{24}\ln^2\!\frac{F}{K} + \dots\right]} \cdot \frac{z}{x(z)} \cdot \left[1 + \left(\frac{(1-\beta)^2 \alpha^2}{24 (FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4 (FK)^{(1-\beta)/2}} + \frac{2 - 3\rho^2}{24}\nu^2\right) T\right] $$

with \(z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln\frac{F}{K}\) and \(x(z) = \ln\frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho}\). Nothing needs to be integrated or simulated, so a fit across a dozen strikes takes a few milliseconds and a whole surface can be re-fitted continuously during the trading day.

Why it works and where it breaks

Three free parameters are enough for most single-maturity smiles, and each has a job: level, tilt, curvature. The expansion is an approximation, though, and it misbehaves for very long maturities, very low strikes and high \(\nu\), where it can even imply a negative density. Modern desks use corrected versions or solve the model's density numerically for those cases. The other limit is that SABR says nothing about how smiles at different maturities relate, which is exactly what Heston and Bates provide.