Hagan's SABR Implied Volatility Asymptotic Expansion

Scenario

The SABR (Stochastic Alpha, Beta, Rho) model is the industry standard for interest rate swaption smile calibration. The forward rate $F_t$ and stochastic volatility $\alpha_t$ follow: $dF_t = \alpha_t F_t^\beta dW_t^1, \quad d\alpha_t = \nu \alpha_t dW_t^2, \quad d\langle W^1, W^2 \rangle_t = \rho dt$

Your Task

1. Implement Hagan et al. (2002)'s singular perturbation expansion for Black implied volatility $\sigma_B(F, K)$ across strikes $K$ and forward $F$.
2. State the analytical formula for both the general case ($F \ne K$) and the at-the-money limit ($F = K$).
3. Compute the implied volatility for strikes $K \in [0.02, 0.04]$ given $F = 0.03, T = 2.0, \alpha = 0.035, \beta = 0.5, \rho = -0.25, \nu = 0.40$, and confirm the volatility smile is convex and strictly positive.

Constraints

• $0 \le \beta \le 1, \nu > 0, |\rho| < 1$.

Examples For ATM strike $K = F$, $\ln(F/K) = 0$, and the formula collapses to the simplified ATM expression.