The monograph provides an exposition of an approach pioneered by the author for pricing contingent claims under processes with jumps. The evolution of the price must be expressed by partial integro-differential equations involving nonlocal terms. Propagating values across distant points can lead to intractable numerical issues that discourage the modeling of jumps. Through a collection of research gathered together in the monograph, the author showed that in some cases the partial integro-differential equation can be transformed into a pseudodifferential equation that does not involve nonlocal terms.
Part II describes the procedure for converting from a partial integro-differential equation to a pseudodifferential equation – making a connection to the characteristic equation of the underlying Lévy process. Part I describes the methods needed to produce a numerical solution. Part III discusses several applications including structural default model with jumps, and local stochastic volatility models with stochastic interest rate and jumps. While the author does not tailor the exposition to practitioners, he presents the material in the style of engineering literature. He aims to make the monograph self-contained through a review of finite difference methods, including Padé approximations and operator splitting, numerical linear algebra including M-matrices and EM-matrices, and Lévy processes.