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<p>In general, it will be difficult to obtain exact analytical solutions to time-fractional Black–Scholes PDEs. Therefore, one has to seek numerical approximate solutions for the same. Since the terminal condition is not smooth, one has to smoothen it, if finite difference schemes are used. To overcome these difficulties we have applied the NIPG method for the first time to solve time-fractional Black–Scholes PDEs. In this approach, the classical <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( L 1-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mn>1</mn> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> scheme is used to discretize the time-fractional derivative on a non-uniform mesh and the spatial derivatives are discretized by the NIPG method on a uniform mesh. We also discuss the L2-stability and convergence results along with the numerical experiments where the numerical results are compared with the exact solutions.</p>

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Numerical Solution of Time-Fractional Black–Scholes PDE by Non-symmetric Interior Penalty Galerkin Method

  • 期刊论文

  • 发表日期：

    2025年2月22日

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  • Jaspreet Kaur,
  • Srinivasan Natesan

摘要

In general, it will be difficult to obtain exact analytical solutions to time-fractional Black–Scholes PDEs. Therefore, one has to seek numerical approximate solutions for the same. Since the terminal condition is not smooth, one has to smoothen it, if finite difference schemes are used. To overcome these difficulties we have applied the NIPG method for the first time to solve time-fractional Black–Scholes PDEs. In this approach, the classical \( L 1-\) L 1 - scheme is used to discretize the time-fractional derivative on a non-uniform mesh and the spatial derivatives are discretized by the NIPG method on a uniform mesh. We also discuss the L2-stability and convergence results along with the numerical experiments where the numerical results are compared with the exact solutions.