Front-fixing FEMs for the pricing of American options based on a PML technique

Abstract

In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods.

Keywords:

• American option pricing
• front-fixing
• perfectly matched layer
• optimal exercise boundary
• discontinuous Galerkin discretization

AMS Subject Classifications:

• 35A35
• 90A09
• 65K10
• 65M12
• 65M60

Acknowledgments

The authors would also like to thank the Centre of High Performance Computing at Jilin University and the Computing Centre of Jilin province, P.R. China.

Additional information

Funding

The work of Kai Zhang was partially supported by the NSF of China [grant number 11271157], the open project program of the state key lab of CAD&CG of Zhejiang University and scientific research foundation for returned scholars, Ministry of Education of China. The work of Jingzhi Li was supported by the NSF of China [grant number 11201453].