Computing Deltas of Callable Libor Exotics in Forward Libor Models

Abstract

Callable Libor exotics is a class of single-currency interest-rate contracts that are Bermuda-style exercisable into underlying contracts consisting of fixed-rate, floating-rate and option legs. The most common callable Libor exotic is a Bermuda swaption. Other, more complicated examples include callable inverse floaters and callable range accruals. Because of their non-trivial dependence on the volatility structure of interest rates, these instruments need a flexible multi-factor model, such as a forward Libor model, for pricing. Only Monte-Carlo based methods are generally available for such models. Being able to obtain risk sensitivities from a model is a prerequisite for its successful application to a given class of products. Computing risk sensitivities in a Monte-Carlo simulation is a difficult task. Monte-Carlo valuation is generally quite slow and noisy. Additionally, numerical noise is amplified when computing risk sensitivities by a "bump-and-revalue" method. Various methods have been proposed to improve accuracy and speed of risk sensitivity calculations in Monte-Carlo for European-type options. Building on previous work in this area, most notably Glasserman and Zhao's (GZ99), we propose a novel extension of some of these methods to the problem of computing deltas of Bermuda-style callable Libor exotics. The method we develop is based on a representation of deltas of a callable Libor exotic as functionals of the optimal exercise time and deltas of the underlying coupons. This representation is obtained by deriving a recursion for the deltas of "nested" Bermuda-style options. The proposed method saves computational effort by computing all deltas at once in the same simulation in which the value is computed. In addition, it produces significantly more stable and less noisy deltas using only a fraction of the number of paths required by the standard "bump and revalue" approach.

Keywords: Bermudan swaptions, callable Libor exotics, callable range accruals, callable inverse floaters, hedging, Greeks, deltas, Monte-Carlo, market model, forward Libor model, Libor market model, LMM, BGM, pathwise deltas