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BIT Numerical Mathematics

, Volume 51, Issue 1, pp 217–233 | Cite as

A multigrid preconditioner for an adaptive Black-Scholes solver

  • Alison RamageEmail author
  • Lina von Sydow
Article

Abstract

This paper is concerned with the efficient solution of the linear systems of equations that arise from an adaptive space-implicit time discretisation of the Black-Scholes equation. These nonsymmetric systems are very large and sparse, so an iterative method will usually be the method of choice. However, such a method may require a large number of iterations to converge, particularly when the timestep used is large (which is often the case towards the end of a simulation which uses adaptive timestepping). An appropriate preconditioner is therefore desirable. In this paper we show that a very simple multigrid algorithm with standard components works well as a preconditioner for these problems. We analyse the eigenvalue spectrum of the multigrid iteration matrix for uniform grid problems and illustrate the method’s efficiency in practice by considering the results of numerical experiments on both uniform grids and those which use adaptivity in space.

Keywords

Finite differences Multigrid Preconditioning Option pricing Adaptive grids 

Mathematics Subject Classification (2000)

65M06 65F08 65M55 

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Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUK
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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