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<p>In this paper, we investigate the stability and convergence of the well-known diagonally implicit Runge-Kutta (DIRK) method, which is applied to solve the European option Black-Scholes equation problem with dividends but without contract costs. The theoretical framework is based on the difference coefficient matrix. It is demonstrated that the DIRK method is stable and convergent in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{L}^{\varvec{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow> <mn mathvariant="bold">2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> norm, provided that the associated difference coefficient matrix is positive definite. Utilizing this sufficient condition, we develop several DIRK methods of up to fourth-order accuracy. These proposed methods will be evaluated through several numerical experiments.</p>

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Stability and convergence analysis of diagonally implicit Runge-Kutta methods for solving black-scholes equations with dividends

  • 期刊论文

  • 发表日期：

    2026年7月14日

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  • Shiyu He,
  • Jincheng Ren

摘要

In this paper, we investigate the stability and convergence of the well-known diagonally implicit Runge-Kutta (DIRK) method, which is applied to solve the European option Black-Scholes equation problem with dividends but without contract costs. The theoretical framework is based on the difference coefficient matrix. It is demonstrated that the DIRK method is stable and convergent in the \(\varvec{L}^{\varvec{2}}\) L 2 norm, provided that the associated difference coefficient matrix is positive definite. Utilizing this sufficient condition, we develop several DIRK methods of up to fourth-order accuracy. These proposed methods will be evaluated through several numerical experiments.