Deriving the Bond Option Payoff of a Payer Swaption
Summary
The document shows how a payer swaption can be expressed as an option on a coupon bond. It starts from a fixed-for-floating swap with reset and payment dates, writes its value as the difference between a floating leg and a fixed leg, and uses the forward Libor definition to simplify the floating-leg cash flows. The resulting swap value is the value of a zero-coupon bond maturing at the swap’s start date minus a coupon bond whose coupon rate equals the swap’s fixed rate.
At expiry, the swaption payoff is the positive part of one minus the value of that coupon bond. This matches a bond option payoff with strike one under the stated unit-notional setup. The derivation provides an algebraic relationship rather than market evidence or a pricing model. Its conclusion relies on the specified swap cash flows, bond definitions, and payment schedule; it does not address adjustments for collateral, alternative conventions, or other swap structures.
Key ideas
- A payer swaption’s underlying swap value can be simplified using forward Libor rates and zero-coupon bond prices.
- The floating leg telescopes to the difference between bonds maturing at the swap’s start and final dates.
- The fixed leg has the cash flows of a coupon bond with coupon rate equal to the swap rate.
- At expiry, the payer swaption payoff is equivalent to a bond option payoff with strike one in the stated setup.
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# Why can a swap option be regarded as a type of Bond option?
# Why can a swap option be regarded as a type of Bond option?
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?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/25630
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.