Computational Economics

, Volume 40, Issue 1, pp 49–62

A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing


DOI: 10.1007/s10614-011-9268-9

Cite this article as:
Cen, Z., Le, A. & Xu, A. Comput Econ (2012) 40: 49. doi:10.1007/s10614-011-9268-9


In this paper we present a stable finite difference scheme on a piecewise uniform mesh along with a power penalty method for solving the American put option problem. By adding a power penalty term the linear complementarity problem arising from pricing American put options is transformed into a nonlinear parabolic partial differential equation. Then a finite difference scheme is proposed to solve the penalized nonlinear PDE, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit time stepping technique. It is proved that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions and is second-order convergent with respect to the spatial variable. Furthermore, our method can efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.


Black–Scholes equation Option valuation Power penalty method Central difference scheme Piecewise uniform mesh 

Mathematics Subject Classification (2000)

65M06 65M12 65M15 

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Institute of MathematicsZhejiang Wanli UniversityNingboPeople’s Republic of China

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