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<p>This work is focused on developing eloquent numerical schemes and their analysis to solve the mixed-type time-fractional Black-Scholes equation (TFBSE), which provides a flexible framework for modeling financial markets with memory effects and multi-scale temporal dynamics. Unlike classical Black-Scholes models, the mixed-type fractional formulation of the Black-Scholes model allows the simultaneous representation of short-term market fluctuations and long-term memory effects frequently observed in asset price dynamics. We first reformulate the mixed-type TFBSE as an equivalent fractional integro-differential equation and construct a scheme based on Crank-Nicolson type discretization. The temporal fractional integral is discretized using piecewise linear interpolation on both uniform and non-uniform meshes, whereas the spatial derivative is approximated using a compact exponential scheme and a Taylor’s compact difference scheme. This leads to a discrete compact scheme that achieves second-order accuracy in time and fourth-order accuracy in space. To further enhance the accuracy, we develop an improved numerical scheme for mixed-type TFBSE that employs the <i>L</i>1-2 method for temporal discretization and a compact difference approach for spatial discretization, ensuring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((3-\alpha)\)</EquationSource> </InlineEquation> accuracy in time and fourth-order accuracy in space. The solvability of the proposed numerical schemes is rigorously demonstrated. Furthermore, the stability and convergence of the proposed schemes are examined through the discrete energy method and Fourier analysis, verifying their robustness and reliability. The numerical experiments are performed to confirm the theoretical convergence orders, second order and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((3-\alpha )\)</EquationSource> </InlineEquation> order in time, and fourth order in space. The numerical errors show that the proposed schemes achieve high accuracy with low computational cost. Further, the numerical results are presented to demonstrate the superior accuracy of the proposed schemes compared to the existing method.</p>

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Novel Computational Methods and Analysis with Temporal Uniform/Non-Uniform Meshes to Solve Mixed-Type Time-Fractional Black-Scholes Equation

  • 期刊论文

  • 发表日期：

    2026年7月29日

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  • Muhammed Fayis P,
  • Suresh Kumar Nadupuri

摘要

This work is focused on developing eloquent numerical schemes and their analysis to solve the mixed-type time-fractional Black-Scholes equation (TFBSE), which provides a flexible framework for modeling financial markets with memory effects and multi-scale temporal dynamics. Unlike classical Black-Scholes models, the mixed-type fractional formulation of the Black-Scholes model allows the simultaneous representation of short-term market fluctuations and long-term memory effects frequently observed in asset price dynamics. We first reformulate the mixed-type TFBSE as an equivalent fractional integro-differential equation and construct a scheme based on Crank-Nicolson type discretization. The temporal fractional integral is discretized using piecewise linear interpolation on both uniform and non-uniform meshes, whereas the spatial derivative is approximated using a compact exponential scheme and a Taylor’s compact difference scheme. This leads to a discrete compact scheme that achieves second-order accuracy in time and fourth-order accuracy in space. To further enhance the accuracy, we develop an improved numerical scheme for mixed-type TFBSE that employs the L1-2 method for temporal discretization and a compact difference approach for spatial discretization, ensuring \((3-\alpha)\) accuracy in time and fourth-order accuracy in space. The solvability of the proposed numerical schemes is rigorously demonstrated. Furthermore, the stability and convergence of the proposed schemes are examined through the discrete energy method and Fourier analysis, verifying their robustness and reliability. The numerical experiments are performed to confirm the theoretical convergence orders, second order and \((3-\alpha )\) order in time, and fourth order in space. The numerical errors show that the proposed schemes achieve high accuracy with low computational cost. Further, the numerical results are presented to demonstrate the superior accuracy of the proposed schemes compared to the existing method.