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?