A class of explicit–implicit alternating parallel difference methods for the two-dimensional Black–Scholes equation

Abstract

The research on the numerical solution of the two-dimensional Black–Scholes equation (the quanto options pricing model) has important theoretical significance and practical value. We propose a class of parallel difference methods for the quanto options pricing model. On the basis of explicit–implicit alternating scheme, inner boundary values of the implicit band are given by explicit calculation of the adjacent points. Then we can get pure alternating band explicit–implicit (PABdE-I) and implicit–explicit (PABdI-E) difference schemes. Numerical experiments and theoretical analysis consistently show that PABdE-I format and PABdI-E format have good parallelism, unconditionally stable and second-order accuracy in time and space. Compared with the classical Crank–Nicolson (C-N) format and alternating band Crank–Nicolson (ABdC-N) format, the calculation efficiency of PABdI-E save 93.56% and 66.13%, in separately. In addition, the calculation accuracy is better than ABdC-N format. The numerical experiments show the schemes given by this paper for solving the quanto options pricing model are high efficiency.

KEYWORDS:

• Two-dimensional Black–Scholes equation
• explicit–implicit alternating parallel difference methods
• stability
• convergence order
• numerical experiments

2000 MR Subject Classifications:

• 65Y05
• 91B24

Acknowledgments

The authors would like to deeply thank Dr. Wu Lifei (School of Mathematics and Physics, North China Electric Power University) for her valuable suggestion and constructive comments during the preparation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Authors contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final version of the manuscript.

Additional information

Funding

The research is supported by the Subproject of Major Science and Technology Program of China (2017ZX07101001-1).