Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION

  • Jeong, Darae (Department of Mathematics Korea University) ;
  • Ha, Taeyoung (Division of Computational Sciences in Mathematics National Institute for Mathematical Sciences) ;
  • Kim, Myoungnyoun (Division of Computational Sciences in Mathematics National Institute for Mathematical Sciences) ;
  • Shin, Jaemin (Institute of Mathematical Sciences Ewha Womans University) ;
  • Yoon, In-Han (Department of Mathematics Korea University) ;
  • Kim, Junseok (Department of Mathematics Korea University)
  • Received : 2013.08.26
  • Published : 2014.07.31
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Abstract

We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

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