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Numerical Approximations of Singular Source Terms in Differential Equations
 J. Comput. Phys
, 2003
"... Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the ..."
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Cited by 25 (1 self)
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Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi dimensions. New high order multi dimensional techniques for di#erential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions to extend one dimensional regularization to higher dimensions may produce O(1) errors.
Convergence remedies for nonsmooth payoffs in option pricing
 J. Computational Finance
"... Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when pricing financial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: ave ..."
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Cited by 17 (1 self)
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Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when pricing financial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: averaging the initial data, shifting the grid, and a projection method. By themselves, these techniques are not sufficient to restore expected behaviour. However, when combined with a special timestepping method, high accuracy is achieved. Examples are provided for one and two factor option pricing problems. 1
Convergence Of Numerical Methods For Valuing PathDependent Options Using Interpolation
, 2002
"... One method for valuing pathdependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially ..."
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Cited by 10 (1 self)
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One method for valuing pathdependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially. We provide a detailed analysis of the convergence of these algorithms. We show that it is possible for the algorithm to be nonconvergent, or to converge to an incorrect answer, if the interpolation scheme is selected inappropriately. We concentrate on Asian options, due to their popularity and because of some errors in the previous literature.
Convergence of spectral and finite difference methods for initialboundary value problems
 SIAM J. Scientific Computing
"... Abstract. The general theory of compatibility conditions for the differentiability of solutions to initialboundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of ..."
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Cited by 6 (3 self)
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Abstract. The general theory of compatibility conditions for the differentiability of solutions to initialboundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of highorder methods. Explicit application of boundary conditions (BCs) that are independent of the initial condition (IC) results in the compatibility conditions not being satisfied. Since this is the case in most science and engineering applications, it is shown that not only does the error in a spectral method, as measured in the maximum norm, converge algebraically, but the accuracy of finite differences is also reduced. For the heat equation with a parabolic IC and Dirichlet BCs, we prove that the Fourier method converges quadratically in the neighborhood of t = 0 and the boundaries and quartically for large t when the firstorder compatibility conditions are violated. For the same problem, the Chebyshev method initially yields quartic convergence and exponential convergence for t>0. In contrast, the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t. These results naturally direct attention to finite difference methods that are also algebraically convergent. In the case of the wave equation, we prove that a secondorder finite difference method is reduced to 4/3order convergence and numerically show that a fourthorder finite difference scheme is apparently reduced to 3/2order. Finally, for the wave equation subject to general ICs and zero BCs, we give a conjecture on the error for a secondorder finite difference scheme, showing that an O(N −2 log N) convergence is possible.
Numerical Methods for Nonlinear Equations in Option Pricing
, 2003
"... This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illust ..."
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Cited by 5 (0 self)
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This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options.
SOME ERROR ESTIMATES FOR THE LUMPED MASS FINITE ELEMENT METHOD FOR A PARABOLIC PROBLEM
"... Abstract. We study spatially semidiscrete lumped masses method for the heat equation with homogeneous Dirichlet conditions. The purpose of this paper is to show that some known smooth initial data estimates for the standard Galerkin method may be carried over to the lumped masses method while nonsmo ..."
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Cited by 1 (1 self)
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Abstract. We study spatially semidiscrete lumped masses method for the heat equation with homogeneous Dirichlet conditions. The purpose of this paper is to show that some known smooth initial data estimates for the standard Galerkin method may be carried over to the lumped masses method while nonsmooth initial data estimates valid under appropriate assumptions on the mesh. 1.
AnnaKarin Tornberg a,*
, 2004
"... www.elsevier.com/locate/jcp Numerical approximations of singular source terms in differential equations ..."
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www.elsevier.com/locate/jcp Numerical approximations of singular source terms in differential equations