Mid curve option replication

Recall mid curve payoff

how we can price it?

Do we know the distribution of Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator.? No. The market has no information.

Can we model Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator.? Yes we can, but we have no idea about its SDE and how to calibrate it.

Replication idea:

Under assumption of the same floating payment structure which is almost always hold,

We can express the forward swap rate Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator. as combination of two swap rates Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator.

The idea is very simple: we substract short swap structure from long swap, so we are left with forward swap structure only. Technically:

Substituting this expression to the standard pricing formula:

Now MCO pricing becomes pricing of function of two swap rates and annuities.