Swaption on Forward-Starting Swap "Replication"?

Question

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 1

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.