Abstract
We propose a new method to efficiently price swap rates derivatives under the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion (DDSVLMM). This method applies series expansion techniques built around Gaussian (Gram-Charlier) or Gaussian mixtures densities to polynomial processes. The standard pricing method for the considered model relies on dynamics freezing to recover a Heston-type model for which analytical formulas are available. This approach is time consuming and efficient approximations based on Gram-Charlier expansions have been recently proposed. In this article, we first discuss the fact that for a class of stochastic volatility model, including the Heston one, the classical sufficient condition ensuring the convergence of Gram-Charlier series is not satisfied. Then, we propose an approximating model based on Jacobi process for which we can prove the stability of Gram-Charlier type expansions. For this approximation, we have been able to prove a strong convergence towards the original model; moreover, we give an estimate of the convergence rate. We also prove a new result on the convergence of the Gram-Charlier series when the volatility factor is not bounded from below. We finally illustrate our convergence results with numerical examples.