Numerical solution of jump-diffusion LIBOR market models
This paper addresses the numerical solution, through discretization and simulation, of the market model with jumps. The model contains special complications arising from the complex form of the intensities describing the dynamics of the marked point processes (MPPs). These intensities are functions of the current LIBOR term structure. At the first step it is shown how to reformulate a term structure model driven by MPPs with suitably bounded state-dependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes. Then the reformulated model is used as a basis for numerical solution, adapting the general approach of \textit{R. Mikulevičius} and \textit{E. Platen} [Math. Nachr. 138, 93-104 (1988; Zbl 0661.60071)]. Briefly, points of the Poisson random measure provide all potential jump times of the LIBOR rates; these may be generated without discretization error in advance of the evolution of the LIBOR rates. In the Poisson jumps the evolution of the LIBOR rates is described by a pure diffusion and may thus be approximated to the desired accuracy using existing methods. Some theoretical support for the method is provided through a result establishing the first and the second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments.
scientific article; zbMATH DE number 2059097
scientific article