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Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates $T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the payments $L(T_{i-1}; T_{i-1}, T_i)\Delta ... 2 it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$\frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A,$$ with$A$the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about$\sigma=0.\$