# Compound option on basket of swaptions

Consider $n$ European Swaptions $S^n$ with exercise dates $T_1 < \dots T_n$. These Swaptions can have different parameters: in particular different strikes, different interest rates, different tenor's and different payment frequencies.

Now consider an option which gives its holder the right but not the obligation to exercise only one of these swaptions at their respective maturity date. To be more precise:

• at $T_1$ the option holder can enter into the first swap. If he does so he may not enter into any other swap
• at $T_k$, given that the option holder did not enter into the first $k-1$ swaps he may choose to enter into the $k$-th swap. If he does so he may not enter into any subsequent swap.

My question is: has this derivative (or similar ones) been studied in the literature before? Given that all swaptions have the same underlying swap this product would equal a Bermudan swaption but in my case the underlying swaps are all different.

Also could such a product be priced via a Longstaff Schwartz algorithm?

• The main difficulties are to properly handle the various projection curves required by non homogeneous libor tenors, and to properly calibrate volatility. You could for instance adapt the Bermudan swaption Hull & White pricer (finite diffs or LS or recursive quadrature) by assuming that the basis between various libor tenors and OIS evolve deterministically along their forwards (the OIS instantaneous rate follows a HW process and all libors are computed from the "OIS libor" and a deterministic basis), and you can then calibrate the time dependent volatility to each of the European swaptions. – Antoine Conze Feb 16 '18 at 14:46