Pricing and hedging callable Libor exotics in forward Libor models

Vladimir V. Piterbarg

Abstract

ABSTRACT

Callable Libor exotics are a class of single-currency interest rate-derivative securities that includes many important types of instruments such as Bermuda swaptions, callable inverse floaters, callable capped floaters, callable range accruals, and the like. These derivatives exhibit complex dependence on the structure of interest rate volatilities, requiring the most sophisticated and flexible models developed so far, forward Libor models, for valuation and risk management. Despite significant practical interest in applications of forward Libor models to callable Libor exotics, a thorough theoretical analysis of problems that arise in such applications has not yet been performed, and this is a gap that is filled in this paper. We present a comprehensive theoretical framework covering the valuation and computation of risk sensitivities. For valuation, the standard Longstaff–Schwartz algorithm for pricing Bermuda swaptions in a Monte Carlo simulation is significantly expanded to include all callable Libor exotics. Importantly, a collection of effective basis functions is constructed. The problem of computing risk sensitivities (Greeks) is given the most attention. A number of new methods for improving the accuracy and computation speed are presented. These include a special Monte Carlo smoothing technique, a very effective control variate method based on a low-dimensional Markovian approximation, and a robust vega computation method.