Numerical valuation of European and American options under Merton's model

Abstract

In this paper, a new combination of time and spatial discretization is proposed for a partial integro-differential equation (PIDE) arising in the valuation of European options under Merton's model. We first present a high-order compact (HOC) difference scheme in space based on a uniform mesh to obtain a highly accurate result, and the discontinuous Galerkin (DG) finite element method in time is introduced that can deal with the loss of the time analyticity. A penalty method is proposed for a partial integro-differential complementarity problem arising in the valuation of the American put option. Numerical experiments are performed to verify the accuracy and efficiency of the proposed method.

Keywords:

• Merton's jump-diffusion model
• European options
• American options
• compact difference scheme
• discontinuous Galerkin finite element

2010 Mathematics Subject Classifications:

• 65M06
• 65M60

Acknowledgements

The authors are thankful to the anonymous referees for reading the manuscript and giving fruitful comments and suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 12071479), and the Outstanding Innovative Talents Cultivation Funded Programs 2019 of Renmin University of China.