# A two-grid penalty method for American options

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## Abstract

In this paper we consider the pricing of American options, governed by a partial differential complementarity problem. The differential problem is first approximated by a semi-linear PDE using two distinct penalty approaches which are well known in computational finance. We then initiate the two-grid algorithm by solving the nonlinear problem on a coarse grid and further the linearized in the interpolated coarse-grid solution problem on a fine grid. By means of the maximum principle the algorithm is shown to be of fourth order convergence rate in space. Numerical experiments verify the presented two-grid approach where we draw some interesting conclusions.

## Keywords

Penalty method Finite difference Two-grid Newton method Maximum principle## Mathematics Subject Classification

35R35 65M06## Notes

### Acknowledgements

Authors would like to thank the anonymous reviewers for their valuable and constructive comments that greatly contributed to improve the quality of the paper.

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