Superconvergence estimates of finite element methods for American options |
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Authors: | Qun Lin Tang Liu Shuhua Zhang |
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Institution: | (1) LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P.R.China;(2) Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, P.R.China |
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Abstract: | In this paper we are concerned with finite element approximations to the evaluation of American options. First, following
W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation
of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is
very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique,
we present sharp L
2-, L
∞-norm error estimates and an H
1-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be
used to generate an efficient a posteriori error estimator.
This work was supported in part by the National Natural Science Foundation of China (10471103 and 10771158), the National
Basic Research Program (2007CB814906), Social Science Foundation of the Ministry of Education of China (Numerical Methods
for Convertible Bonds, 06JA630047), Tianjin Natural Science Foundation (07JCY-BJC14300), and Tianjin University of Finance
and Economics. |
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Keywords: | American options variational inequality finite element methods optimal and superconvergent estimates interpolation postprocessing a posteriori error estimators |
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