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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 ...


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 $\sigma=0.$


Exploiting an arbitrage is straightforward. Constructing and noticing one is the hard part. In your case if you know that Swptn(K,T1,T2)+Swptn(K,T2,T3) >= Swptn(K,T1,T3), Simply sell Swptn(K,T1,T2)+Swptn(K,T2,T3) and buy Swptn(K,T1,T3). Sell the most expensive and buy the cheapest. L.

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