An Extended Libor Market Model With Nested Stochastic Volatility Dynamics

In this paper we extend standard Libor Market Model (LMM) with nested stochastic volatilities. The stochastic volatility of each Libor follows a mean-reverting process as in Schoebel and Zhu (1999) or in Heston (1993) under the individual forward measure of each Libor. Other than the existing stochastic volatility models, every volatility in the extended LMM is correlated with its Libor individually, and the parameters of stochastic volatility are also different over all Libors, however, are nested by some deterministic functions. With a nesting function, the same type of parameter such as mean level in all volatility processes share a certain term structure. In this model set-up, we can still derive the stochastic processes for Libors and volatilities under an arbitrary forward measure. In line with the stochastic volatility models for equity options, we obtain a closed-form solution via Fourier transform for caplets and floorlets. Finally, we use factor representation to express Libors and swap rates by some independent factors, namely principle components. The approximated analytical pricing formula principal for swaption can then be derived by using the characteristic functions that are just a product of the characteristic function of each factor. The numerical implementation of the nested stochastic volatility model is efficient and identical to the existing stochastic volatility models