 
Summary: A STOCHASTIC VOLATILITY MODEL FOR BERMUDA SWAPTIONS AND
CALLABLE CMS SWAPS
CLAUDIO ALBANESE AND MANLIO TROVATO
Abstract. It is widely recognized that fixed income exotics should be priced by means of
a stochastic volatility model. Callable constant maturity swaps (CMS) are a particularly
interesting case due to the sensitivity of swap rates to implied swaption volatilities for very
deep out of the money strikes. In this paper, we introduce a stochastic volatility term structure
model based on a continuous time lattice which allows for a numerically stable and quite
efficient methodology to price fixed income exotics in this class.
1. Introduction
The history of interest rate models is characterized by a long series of turns. The Black
formula for caplets and swaptions was designed to take as underlying a single forward rate
under the appropriate forward measure. This approach has the advantage to lead to a simple
pricing formula for European options but also the limitation of not being extendable to callable
contracts. To have a more consistent model, short rate models where introduced in (Cox,
Ingersoll & Ross 1985), (Vasicek 1977), (Black & Karasinski 1991) and (Hull & White 1993).
Next came LIBOR market models, also known as correlation models. First introduced in (Brace,
Gatarek & Musiela 1996) and (Jamshidian 1997), forward LIBOR models affirmed themselves as
a flexible Montecarlo pricing methodology providing nonparametric fits to both term structures
for interest rates and atthemoney Black volatilities for either caplets or a family of swaptions of
