Efficient numerical pricing of American options based on multiple shooting method: a PDE approach

ABSTRACT

In this paper, the Black–Scholes (B-S) equation to price American options is studied which is governed by a partial differential problem. A nonlinear partial differential equation (PDE) is resulted by applying a penalty approach for this problem. To numerically solve this PDE, an equipped finite difference method with variable step size (VSS) in the time direction is designed and implemented, coupled with a boundary value solver based on a multiple shooting method (MSM) in the spatial direction. It is completely proven that the proposed approximate option prices satisfy the early exercise constraint. Moreover, the convergence of the proposed method is analysed. The extracted numerical results indicate the punctuality and effectiveness of the present scheme comparing the previous works.

KEYWORDS:

• American option pricing
• Black–Scholes equation
• multiple shooting method
• partial differential equation
• penalty method
• Richardson extrapolated method

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 41A55
• 91G20

Acknowledgments

The authors like to thank the associate editor and anonymous referees for their comments that have improved the structure of this paper.

Disclosure statement

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

Additional information

Funding

The work of first author was supported by the University of Tabriz, Iran under Grant No. [3902].