Bermudan Swaptions in the LIBOR Market Model

Abstract

Bermudan swaptions have until recently been valued using only one-factor models such as the Black-Derman-Toy (BDT) or Black-Karasinski (BK) models. The LIBOR Market (LM) model which is a more general multi-factor model is becoming increasingly popular as a benchmark model. Whereas the BDT and BK models can be approximated using a lattice facilitating easy valuation of Bermudan swaption, the LM model doesn't conform to the lattice framework and as such the valuation seems very difficult. Monte-Carlo simulation is a popular alternative to the lattice framework for derivatives valution. In order to facilitate valuation of Bermudan swaptions the Monte-Carlo simulation technique must be extended. A few methods doing this are presently available, eg [And98]. A common feature of these methods is that the estimated option premia are only lower bounds on the true premia. The Stochastic Mesh method proposed by [BG97b] for valuation of Bermudan (equity) options with applications to equity options provides a lower and an upper bound. We have applied this method to the LM model and use this to verify the premia found by Andersen. We will also apply the approach suggested in [LS98] to the LM model and verify the premia found using that approach. As it turns out this approach is a special case of the [And98] approach. Furthermore we also examine the impact on the Bermudan swaption premia when moving from a LM model with only one factor to a LM model with multiple factors and do indeed find a significant---but not dramatic---impact. We find the [And98] and [LS98] approaches to be mutually consistent and in line with results obtained from low-biased Stochastic Mesh estimates.