Smiling at convexity

The price of a constant maturity swap (CMS)-based derivative is largely determined by the value of swaption volatilities at extreme strikes. Fabio Mercurio and Andrea Pallavicini propose a simple procedure for stripping consistently implied volatilities and CMS adjustments from the market quotes of swaption smiles and CMS swap spreads

Information on implied volatilities of swap rates is provided by the market both directly through quotes of swaption smiles and indirectly through prices of constant maturity swaps (CMSs), where a CMS rate is exchanged for a Libor rate plus a spread (referred to as a CMS spread).

Unfortunately, not every swaption in the standard at-the-money matrix also has quotes for away-from-the-money strikes, and when present, the latter are often too few to allow for a robust bootstrap of the whole smile