Why Forward-Starting Swaptions Depend on a Spread of Swap Rates
Summary
The document examines whether a swaption expiring at an intermediate date and entering a later-starting swap can be replicated by combining two swaptions. The accepted explanation rewrites the underlying forward-starting swap as a long swap from the intermediate date to the final date and a short swap from that date to an earlier endpoint. At expiry, the payer option is therefore on the difference between two swap values, each formed from its swap rate and annuity.
This makes the payoff a weighted spread option whose value depends on the joint distribution of the two swap rates. The prices of separate vanilla swaptions do not by themselves capture that dependence, so the proposed combination cannot perfectly replicate the target payoff. The response allows that bounds or approximations may be possible, but supplies no calibration procedure, numerical example, or specific bound. Its conclusion depends on correctly matching swap schedules and annuities to the contract.
Key ideas
- A forward-starting swap can be represented as a long swap over the full period and a short swap over the initial subperiod.
- The payer swaption payoff is an option on the difference between the two swap values.
- Each swap value depends on both its swap rate and its annuity.
- The option value depends on the joint distribution of the two swap rates.
- Vanilla swaption prices alone do not provide an exact replication, though bounds or approximations may be available.
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# Swaption on Forward-Starting Swap "Replication"?
# Swaption on Forward-Starting Swap "Replication"?
Lately I was thinking about forward-starting swaptions vs. options on forward-starting swaps a bit, and I started wondering about the following:
Suppose we are at time $T_0$ (today) and we want to price a swaption that expires in $T_1$ and entitles us to enter into a swap which lives from $T_2$ to $T_3$. Clearly, I work in the setting $T_0 < T_1 < T_2 < T_3$.
I was asking myself whether it is reasonable (possible?) to approximate (replicate?) the price of above mentioned option by looking at a combination of the prices of:
- a spot ($T_0$) starting swaption with expiry $T_2$ that delivers the (then, i.e., at $T_2$) spot-starting swap and
- a forward-starting swaption that lives from $T_1$ to $T_2$ and delivers the (then, i.e., at $T_2$) spot-starting swap
I have drawn a little picture to illustrate what I mean ($T_0=0$ (today), $T_1$ is 1 year from today, $T_2$ is 3 years from today, and $T_3$ is 6 years from today):
I intuitively have the feeling that it's not working out, and my first line of thought is that it's because the swap underlying the three options is not 100% the same (although it's always the 3x6 swap, the forward starting swap seems more uncertain to me compared to the then-spot starting swap, as the optionality ends after 1y and not after 3y). Maybe someone can provide a little more information and/or some formulae that would confirm my conjecture?
## Answer by Marco (score 2, accepted)
https://quant.stackexchange.com/a/60057
The way to think about this is an option to enter a basket of two swaps. The basket contains these positions:
$P_1$: a long position in a swap that starts at $T_1$ and finishes at $T_3$
$P_2$: a short position in a swap that starts at $T_1$ and finishes at $T_2$.
This basket replicates the payoff of the forward starting swap. Denoting $S(\tau_1, \tau_2)$ as the swap rate for the swap starting at $\tau_1$ and ending at $\tau_2$, and $A(\tau_1, \tau_2)$ as the corresponding Annuity (PVBP), then the payoff (for a payer) can be written as:
\begin{equation} \max \left \{ \underbrace{A(T_1, T_3) (S(T_1,T_3)-K)}_{P_1} - \underbrace{A(T_1,T_2) (S(T_1,T_2)-K)}_{P_2}, 0\right \} \end{equation}
This is effectively a spread option between two swap rates (obviously with some weights). The present value of the spread option therefore depends on the joint distribution between the two swap rates, $S(T_1, T_2)$ and $S(T_1, T_3)$. So you will not be able to perfectly replicate this payoff with vanilla swaptions, though some (upper / lower bound) approximations may be possible.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.