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<p>This paper presents a computational algorithm based on the collocation method to solve a time-fractional partial integro-differential equation (FPIDE) for option pricing. We begin with a long-memory stochastic financial model incorporating jumps and derive a partial integro-differential equation (PIDE) for European options using risk-neutral valuation and self-financing portfolio strategies. To enhance realism, we extend this model to a time-fractional PIDE. To address the non-smooth solution of the FPIDE, we introduce fractional basis functions derived from Morgan-Voyce polynomials. The solution is expressed as a double series expansion using these fractional basis functions. We then compute an operational matrix to approximate the fractional and partial derivatives in the FPIDE. The integral term is handled using the Hermite quadrature rule. Applying the collocation method, we transform the problem into a linear system of equations. A detailed theoretical analysis proves the convergence of our method. To validate its efficiency, we test four cases: two with smooth exact solutions, one with a non-smooth exact solution, and one without a known exact solution. The results demonstrate the effectiveness of our approach in solving fractional option pricing problems.</p>

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Collocation method with Morgan-Voyce polynomials to solve the time fractional long memory Black-Scholes model with jump process

  • 期刊论文

  • 发表日期：

    2025年7月30日

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  • Panumart Sawangtong,
  • Alireza Najafi

摘要

This paper presents a computational algorithm based on the collocation method to solve a time-fractional partial integro-differential equation (FPIDE) for option pricing. We begin with a long-memory stochastic financial model incorporating jumps and derive a partial integro-differential equation (PIDE) for European options using risk-neutral valuation and self-financing portfolio strategies. To enhance realism, we extend this model to a time-fractional PIDE. To address the non-smooth solution of the FPIDE, we introduce fractional basis functions derived from Morgan-Voyce polynomials. The solution is expressed as a double series expansion using these fractional basis functions. We then compute an operational matrix to approximate the fractional and partial derivatives in the FPIDE. The integral term is handled using the Hermite quadrature rule. Applying the collocation method, we transform the problem into a linear system of equations. A detailed theoretical analysis proves the convergence of our method. To validate its efficiency, we test four cases: two with smooth exact solutions, one with a non-smooth exact solution, and one without a known exact solution. The results demonstrate the effectiveness of our approach in solving fractional option pricing problems.