Weak Galerkin finite element method for valuation of American options  

作　　者：Ran ZHANG Haiming SONG Nana LUAN 

机构地区：[1]School of Mathematics, Jilin University, Changchun 130012, China [2]School of Insurance and Economics, University of International Business and Economics, Beijing 100190, China

出　　处：《Frontiers of Mathematics in China》2014年第2期455-476,共22页中国高等学校学术文摘·数学（英文）

基　　金：Acknowledgements The authors would like to thank the anonymous referees for their careflll reading of the manuscript and their valuable comments. The authors also wish to thank the High Performance Computing Center of Jilin University and C, omputing Center of ,lilin Province for essential support. This work was supported by the National Natural Science Foundation of China Grant Nos. 11271157, 11371171）, the Open Project Program of the State Key Lab of CAD＆CG （A1302） of Zhejiang University, the Scientific Research Foundation for bleturned Scholars, Ministry of Education of China. and UIBE （11QD17）.

摘　　要：We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.

关 键 词：American option optimal exercise boundary weak Galerkin finite element method 

分 类 号：O241.82[理学—计算数学] F830.9[理学—数学]