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?

  • $\begingroup$ how would you write $P(fl)$ as a function of $fl$ ? If $P(fl)=\frac{1}{1+fl(T-t)}$ then you get : $\frac{T-t}{(1+fl(T-t))^2}(fl-s)^+\leq (P(fl)-P(s))^+ \leq (T-t)(fl-s)^+$ $\endgroup$ Commented Apr 25, 2016 at 12:01

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.