Penalty methods for the numerical solution of American multi-asset option problems

Authors:
Bjørn Fredrik Nielsen;Ola Skavhaug;Aslak Tveito
Affiliations:
Simula Research Laboratory, P.O. Box 134, N-1325 Lysaker, Norway and Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316 Oslo, Norway;Simula Research Laboratory, P.O. Box 134, N-1325 Lysaker, Norway and Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316 Oslo, Norway;Simula Research Laboratory, P.O. Box 134, N-1325 Lysaker, Norway and Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316 Oslo, Norway
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.