Volatility Dynamics in a Term Structure

In this chapter we start by considering a generalised framework, encompassing in particular the Caplets and Swaptions markets, but potentially applicable to other products. This is made possible because these payoffs, as well as the martingale method used to price them, are very similar. Hence the main requirement is to find the correct numeraire and pricing measure. The difference with the single underlying setting of Part I is that we are now dealing with a collection of underlyings, for instance the forward Libor or forward par swap rates. Each of these underlying has its own numeraire, is martingale under the associated measure, and defines a specific strike-continuum of vanilla options. Hence we end up with associated collections of numeraires, measures and options. All these families are parametrised by their own list of maturities, which we will naturally extend to a common maturity continuum. We end up naturally with a term structure (TS) framework, and in solving the direct and indirect problems we will point to the structural difference simpler single-underlying environment of Part I.