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Multidimensional Lévy processes, Numerical methods, Asset pricing, 60J75, 65N06, 65N30, C63,

In this work we present and discuss a possible globalization concept for Newton-type methods. We consider nonlinear problems f (x) = 0 in R n using the concepts from ordinary differential equations as a basis for the proposed numerical solution procedure. Thus, the starting point of our approach is within the framework of solving ordinary differential equations numerically. Accordingly, we are able to reformulate general Newton-type iteration schemes using an adaptive step size control procedure. In doing so, we derive and discuss a discrete adaptive solution scheme, thereby trying to mimic the underlying continuous problem numerically without losing the famous quadratic convergence regime of the classical Newton method in a vicinity of a regular solution. The derivation of the proposed adaptive iteration scheme relies on a simple orthogonal projection argument taking into account that, sufficiently close to regular solutions, the vector field corresponding to the Newton scheme is approximately linear. We test and exemplify our adaptive root-finding scheme using a few low-dimensional examples. Based on the presented examples, we finally show some performance data.

Parameter sensitivities of prices for derivative contracts play an important role in model calibration as well as in quantification of model risk. In this paper a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented. Variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model differentiated with respect to the model parameters are employed. Superconvergent approximations to second and higher derivatives of prices with respect to the price process' state variables are extracted from approximate, computed prices with low, C 0 regularity by postprocessing. The extracted numerical sensitivities are proved to converge with optimal rates as the mesh width tends to zero. Numerical experiments for uni-and multivariate models with sparse tensor product discretization confirm the theoretical results.

In this article, we provide an overview of wavelet methods for asset pricing. Wavelet methods are a particular realization of the finite element method. They provide a very general PDE‐based numerical pricing technique. We describe the general setup and admissible market models. The fundamental concepts of the wavelet‐based finite element implementation are described. Furthermore, we illustrate the pricing of American contracts as well as the computation of sensitivities and Greeks. Numerical illustrations are given.

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