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An Implicit Scheme for American Put Options

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Abstract

In this paper, an implicit scheme is proposed to solve a parabolic variational inequality arising from the American put options. The discretization leads to a class of discrete elliptic variational inequalities. Well-posedness, including existence, uniqueness, comparison principle, and stability of the discrete elliptic variational inequality is established. A simple and efficient algorithm to solve the implicit discretized variational inequality is discovered. The novelty here is an explicit formula for the optimal exercise boundary. An improved algorithm is also presented to eliminate the singularity near the time to expiry. Numerical examples are carried out to show the accuracy and efficiency of the proposed algorithms.

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Acknowledgements

The third author was supported in part by the National Natural Science Foundation of China through Grant [12071373] and the last author was supported in part by National Natural Science Foundation of China (Grant No. 12101509) and was supported in part by the Fundamental Research Funds for the Central Universities (JBK2304085). The authors are very grateful to the anonymous referees for many constructive comments and suggestions which have improved this paper.

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  1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Xinfu Chen, Zhengyang Lu, Jingtang Ma & Jinye Shen

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  1. Xinfu Chen
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Correspondence to Jingtang Ma or Jinye Shen.

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Chen, X., Lu, Z., Ma, J. et al. An Implicit Scheme for American Put Options. J Sci Comput 97, 42 (2023). https://doi.org/10.1007/s10915-023-02356-6

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  • DOI: https://doi.org/10.1007/s10915-023-02356-6

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