SABR Implied Volatility (Hagan)

A swaptions desk gets a request to quote a strike 50 basis points out of the money, with no direct market print to lean on. That gap is what this calculator fills. It runs Hagan's 2002 closed-form approximation to the SABR model and returns the Black/lognormal implied volatility for the forward and strike you enter. Rates and FX vol traders reach for it to interpolate the smile from a handful of calibrated points and keep neighboring strikes consistent, without firing off a Monte Carlo run for every quote.

SABR models the forward as a CEV diffusion with elasticity beta, whose volatility alpha is itself stochastic — driven by the vol-of-vol nu and tied to the forward through the correlation rho. Hagan and his coauthors attacked that system with singular perturbation theory and produced an analytic expression for the Black vol, built around the variable z = (nu/alpha)·(F·K)^((1−beta)/2)·ln(F/K) and a function x(z) that encodes the correlation skew. Keep one caveat in mind: this is an asymptotic expansion. At long maturities or very low strikes it loses accuracy and can imply negative densities in the wings — the well-known SABR arbitrage problem that later prompted the PDE and Antonov refinements.

Enter the forward F, the strike K, the time to expiry in years, and the four calibrated parameters: alpha (initial vol), beta (CEV elasticity), rho (correlation), and nu (vol of vol). The output is the lognormal volatility you drop straight into Black-76 to price the option. To sanity-check it, set K = F: the result should collapse to the at-the-money level, near alpha/F^(1−beta). Rho sets the sign of the skew — negative tilts it toward low strikes — while nu drives the curvature, so a larger nu bends the smile into a deeper convex shape.