18 research outputs found

    Numerical methods for Lévy processes

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    We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model

    On Kolmogorov equations for anisotropic multivariate Lévy processes

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    For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Singularity-free representations of the Dirichlet forms are given which remain bounded for piecewise polynomial, continuous functions of finite element type. We prove that the variational problem can be localized to a bounded domain with explicit localization error bounds. Furthermore, we collect several analytical tools for further numerical analysi

    Compression of boundary integral operators discretized by anisotropic wavelet bases

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    The present article is devoted to wavelet matrix compression for boundary integral equations when using anisotropic wavelet bases for the discretization. We provide a compression scheme which amounts to only O(N)O(N) relevant matrix coefficients in the system matrix without deteriorating the accuracy offered by the underlying Galerkin scheme. Here, NN denotes the degrees of freedom in the related trial spaces. By numerical results we validate our theoretical findings

    A fast sparse grid based space-time boundary element method for the nonstationary heat equation

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    This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin's method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the H -matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

    Option Pricing under Multifactor Black-Scholes Model Using Orthogonal Spline Wavelets

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    The paper focuses on pricing European-style options on several underlying assets under the Black-Scholes model represented by a nonstationary partial differential equation. The proposed method combines the Galerkin method with L2L^2-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping. To this end, we construct an orthogonal cubic spline wavelet basis on the interval satisfying homogeneous Dirichlet boundary conditions and design a wavelet basis on the unit cube using the sparse tensor product. The method brings the following advantages. First, the number of basis functions is significantly smaller than for the full grid, which makes it possible to overcome the so-called curse of dimensionality. Second, some matrices involved in the computation are identity matrices, which significantly simplifies and streamlines the algorithm, especially in higher dimensions. Further, we prove that discretization matrices have uniformly bounded condition numbers, even without preconditioning, and that the condition numbers do not depend on the dimension of the problem. Due to the use of cubic spline wavelets, the method is higher-order convergent. Numerical experiments are presented for options on the geometric average.Comment: 43 pages, 10 figure

    The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation

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    In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
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