ABSTRACT
This chapter is devoted to analyzing swaprate dynamics, finding suitable deterministic approximations to swaprate volatilities, and deriving corresponding swaption formulae. In the shifted-BGM model constructed in the last chapter the forwards
plus time independent shifts were dynamically lognormal under the appropriate forward measures. We now show, following [31] (and others - it’s the sort of result many implementers have probably obtained independently) that swaprates in this framework exhibit the same sort of behavior as forwards in that swaprates plus shifts that are nearly time independent are dynamically almost lognormal under new measures, equivalent to the forward measures, that we call swaprate measures. Thus in the first Section-4.1 we separate the swaprate into its shift and sto-
chastic parts, and then in Section-4.2 and Section-4.3, analyze each separately and justify our approximations. A consequence obtained in Section-4.4, is that in shifted-BGM swaptions
can be priced accurately with Black type formulae very similar to the ones (3.5) used to price caplets. This is a standard outcome in the BGM framework - whatever method prices caplets, also usually fairly accurately prices swaptions after some necessary adjustments and approximations. For example, see Chapter-16 on the stochastic volatility version of BGM for a similar outcome, or [9] and [95] for generic methods using Markovian projection. When the shift in BGM is zero the swaprate SDEs (4.13) permit, see
Section-4.5, the easy derivation of Jamshidian’s [66] swaprate model in which the swaprates of coterminal swaps can be made jointly lognormal under a collection of appropriate swaprate measures (rather similar to the forwards under the collection of forward measures in BGM). But more than that, it’s possible to construct many other market models in which the swaprates of any set of swaps with a strictly increasing total tenor structure, which may include forwards, can be made jointly lognormal under appropriate measures. When the shift is non-zero, however, further work is needed to properly
construct a Jamshidian type swaprate model that behaves like the forwards in shifted BGM with an exact closed formula for swaptions corresponding to (3.5) for caplets. Among all the market models, the relative algebraic simplicity of shifted-
BGM combined with its ability to handle swap dynamics and calibrate to
a model.
