A Finite Difference Method for a Normalized Time-Fractional Black–Scholes Equation

초록

In this article, we propose a normalized time-fractional Black–Scholes (TFBS) equation. The proposed model uses a normalized time-fractional derivative which has a distinctive feature wherein a weight function possesses the property that its integral with respect to time is always equal to one. This feature ensures a well-balanced integration over time and provides a fair comparison between different fractional orders. An implicit finite difference method is used for the numerical solution of the TFBS equation. We perform several standard numerical tests to examine the effect of the fractional parameter on the option pricing. These experiments are designed to investigate how variations in the fractional parameter influence pricing outcomes and provide valuable insights into the model’s behavior under different conditions. The computational results demonstrate the model’s potential for capturing more intricate market dynamics, which makes the proposed model a promising tool for financial analysis. In the appendix, we provide a computer program that implements the numerical methods discussed in the paper for interested readers. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.

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