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Why can a swap option be regarded as a type of bond option?
My idea: Suppose the swap rate of the swaption is $s$. Now consider a bond option expiring at $T$ with strike, $(P_K)_t = \dfrac{1}{1+s(T-t)}$. The bond payoff is given by $(P(fl)-P_K)_+$ and the swaption payoff is given as $(fl-s)_+$. There is clearly a relationship between these two payoffs.
Can someone provide a mathematical proof of the result?
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 $L(T_{i-1}; T_{i-1}, T_i)\Delta T_i$ and $K\Delta T_i$, where $\Delta T_i = T_i -T_{i-1}$, and \begin{align*} L(t; T_{i-1}, T_i) = \frac{1}{\Delta T_i}\bigg(\frac{P(t, T_{i-1})}{P(t, T_i)}-1 \bigg), \end{align*} for $i=1, \ldots, n$, is a forward Libor rate. Here, $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face value.
The value of the swap at time $t$, where $0 \leq t \le T_0$, is given by \begin{align*} & \ \sum_{i=1}^n \frac{1}{\Delta T_i}\bigg(\frac{P(t, T_{i-1})}{P(t, T_i)}-1 \bigg) \times \Delta T_i \times P(t, T_i) - K \sum_{i=1}^n P(t, T_i) \times \Delta T_i \\ = & \ P(t, T_0)-P(t, T_n) - K \sum_{i=1}^n P(t, T_i) \Delta T_i\\ = & \ P(t, T_0)- \bigg(\sum_{i=1}^{n-1}K \Delta T_i P(t, T_i) + \big( 1+ K \Delta T_n\big) P(t, T_n) \bigg). \end{align*}
The swaption payoff at maturity $T_0$ is given by \begin{align*} & \ \Bigg[\sum_{i=1}^n \frac{1}{\Delta T_i}\bigg(\frac{P(T, T_{i-1})}{P(T, T_i)}-1 \bigg) \times \Delta T_i \times P(T, T_i) - K \sum_{i=1}^n P(T, T_i) \times \Delta T_i\Bigg]^+ \\ = & \ \Bigg[1- \bigg(\sum_{i=1}^{n-1}K \Delta T_i P(T_0, T_i) + \big( 1+ K \Delta T_n\big) P(T_0, T_n) \bigg)\Bigg]^+. \end{align*} That is, the swaption payoff is the payoff of a bond option that has a coupon rate the same as the swap fixed rate, while the bond option strike is 1.
