The SABR Model

SABR models the forward price directly with four parameters: alpha sets the initial volatility level, beta fixes the backbone linking volatility to the forward, rho drives the skew through correlation, and nu, the vol of vol, controls smile curvature. Hagan’s asymptotic expansion converts these into an explicit implied-volatility formula with instant quotes at any strike, which made SABR the default quoting convention for rates desks trading swaptions and for many FX options markets. It remains an expansion, however: far from the money it can imply negative densities and arbitrage, so careful practitioners restrict it to its valid range or repair the wings with more robust parameterizations.