Journal of the Korean Society for Industrial and Applied Mathematics

  • 제28권3호
  • /
  • Pages.88-95
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  • 2024
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
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)
  • 투고 : 2024.07.18
  • 심사 : 2024.09.24
  • 발행 : 2024.09.25
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초록

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.

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과제정보

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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