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A second-order finite difference method for the Black–Scholes model without far-field boundary conditions

Author

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  • Wang, Jian
  • Wu, Lin
  • Wu, Xinpei
  • Hwang, Youngjin
  • Nam, Yunjae
  • Kwak, Soobin
  • Lee, Taehui
  • Kim, Junseok

Abstract

We propose an explicit finite difference method for the Black–Scholes (BS) equation that avoids artificial far-field boundary conditions. The method uses an alternating direction explicit (ADE) update on a dynamically shrinking grid, thereby eliminating the need for boundary values at the far end of the domain. It effectively alleviates the stability constraints of the explicit format through the alternating direction advancement. Numerical experiments on European and cash or nothing options confirm second-order convergence and demonstrate a high level of efficiency. For example, repeatedly doubling time resolution from 160 to 1280 reduces pricing error from 7.45×10−1 to 6.56×10−3, with observed convergence rates close to 2. This makes the method suitable for low-latency financial applications such as real-time pricing and risk management.

Suggested Citation

  • Wang, Jian & Wu, Lin & Wu, Xinpei & Hwang, Youngjin & Nam, Yunjae & Kwak, Soobin & Lee, Taehui & Kim, Junseok, 2025. "A second-order finite difference method for the Black–Scholes model without far-field boundary conditions," Journal of Financial Stability, Elsevier, vol. 81(C).
    • Handle: RePEc:eee:finsta:v:81:y:2025:i:c:s1572308925001068
    DOI: 10.1016/j.jfs.2025.101477
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    References listed on IDEAS

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