Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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AN EFFICIENT AND ACCURATE ADAPTIVE TIME-STEPPING METHOD FOR THE BLACK-SCHOLES EQUATIONS

  • HYEONGSEOK HWANG (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • SOOBIN KWAK (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • YUNJAE NAM (PROGRAM IN ACTUARIAL SCIENCE AND FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • SEOKJUN HAM (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • ZHENGANG LI (PROGRAM IN ACTUARIAL SCIENCE AND FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • HYUNDONG KIM (DEPARTMENT OF MATHEMATICS AND PHYSICS, GANGNEUNG-WONJU NATIONAL UNIVERSITY) ;
  • JUNSEOK KIM (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY)
  • Received : 2024.07.18
  • Accepted : 2024.09.24
  • Published : 2024.09.25
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Abstract

In this article, we propose an efficient and accurate adaptive time-stepping numerical method for the Black-Scholes (BS) equations. The numerical scheme used is the finite difference method (FDM). The proposed adaptive time-stepping computational scheme is based on the maximum norm of the discrete Laplacian values of option values on a discrete domain. Most numerical solvers for the BS equations require a small time step when there are large variations in the solutions. To resolve this problem, we propose an adaptive time-stepping algorithm that uses a small time step size when the maximum norm of the discrete Laplacian values on a discrete domain is large; otherwise, a larger time step size is used to speed up the computation. To demonstrate the high performance of the proposed adaptive time-stepping methodology, we conduct several computational experiments. The numerical tests confirm that the proposed adaptive time-stepping method improves both the efficiency and accuracy of computations for the BS equations.

Keywords

Acknowledgement

The corresponding author (J.S. Kim) was supported by the Brain Korea 21 FOUR through the National Research Foundation of Korea funded by the Ministry of Education of Korea.

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