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Mathematical analysis and numerical methods for a partial differential equations model governing a ratchet cap pricing in the LIBOR market model. (English) Zbl 1222.91061
Summary: We present a mathematical model for pricing a particular financial product: the ratchet cap. This derivative product depends on certain interest rates (whose dynamics we assume that follow the LIBOR market model). The pricing model is rigorously posed in terms of a sequence of nested Cauchy problems associated to uniformly parabolic partial differential equations. First, for each problem the existence and uniqueness of solution is obtained. Next, this analysis allows to propose a new and more efficient numerical method based on the approximation by computable fundamental solutions of constant coefficient operators. The advantage in terms of computational time of this new modeling and analytically based approach is illustrated by comparison with the classically used Monte Carlo simulation and a characteristics Crank-Nicolson time discretization combined with finite elements strategy.
MSC:
91G20Derivative securities
91G30Interest rates (stochastic models)
35A35Theoretical approximation to solutions of PDE
35K15Second order parabolic equations, initial value problems
65M25Method of characteristics (IVP of PDE, numerical methods)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35A08Fundamental solutions of PDE