Libor Market Model with Stochastic Volatilities

In this chapter we address the smile modeling with stochastic volatility within the setup of Libor Market Model (LMM). Meanwhile, LMM is a standard pricing model for interest rate derivatives. Since LMM is based on a series of lognormal dynamics, the methods for building up the smile models, particularly with stochastic volatilities, can be adopted from the previous chapters. After a brief introduction to interest derivative markets in Section 11.1, we review in Section 11.2 a standard LMM with which we may gain the first impression of the high dimensionality and the large dependence structure of LMM. Next in Section 11.3, swaption valuation and Swap Market Model (SMM) are explained. The consistent valuation of swaptions and swap-rate linked products (CMS products) with LMM remains a challenge in quantitative finance. The key section in this chapter is Section 11.4 where five stochastic volatility LMMs are discussed. All of these five models apply characteristic functions or moment-generating functions for pricing caps and swaptions to different extent. While CFs do not find significant applications in the models of Andersen and Brotherton-Ratcliffe (2001), and Piterbarg (2003), the models of Zhang and Wu (2006), Zhu (2007) as well as Belomestny, Matthew and Schoenmakers (2007) have used CFs extensively and consequently to arrive at the closed-form pricing formulas for caplets and swaptions. The last section provides some conclusive remarks.