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Superconvergence estimates of finite element methods for American options

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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.

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P.R.China

    Qun Lin

  2. Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, P.R.China

    Tang Liu & Shuhua Zhang

Authors
  1. Qun Lin
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  2. Tang Liu
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  3. Shuhua Zhang
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Corresponding author

Correspondence to Shuhua Zhang.

Additional information

Dedicated to Ivan Hlaváček on the occasion of his 75th birthday

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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Lin, Q., Liu, T. & Zhang, S. Superconvergence estimates of finite element methods for American options. Appl Math 54, 181–202 (2009). https://doi.org/10.1007/s10492-009-0012-x

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