Mathematical analysis and numerical methods for a PDE model governing a ratchet-cap pricing in the Libor Market Model

In this paper we present a PDE formulation for the ratchet cap pricing problem. The underlying LIBOR interest rates are assumed to follow the LIBOR market model. For this PDE problem the existence and uniqueness of solution are obtained in the classical framework of uniformly parabolic PDEs in terms of a sequence of nested Cauchy problems. Moreover, this approach allows to obtain a new numerical method based on the approximation by computable fundamental solutions of constant coefficient operators. This method is compared with classical Monte Carlo simulation and a proposed characteristics Crank-Nicolson time discretization combined with finite elements strategy.