Lie symmetry analysis for the fractal Black-Scholes equation of option pricing

Yuan Yue; Chao Yue; Yuan Yue; Chao Yue

doi:10.3934/math.20251036

AIMS Mathematics

2025, Volume 10, Issue 10: 23346-23359. doi: 10.3934/math.20251036

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Research article

Lie symmetry analysis for the fractal Black-Scholes equation of option pricing

Yuan Yue ¹,
Chao Yue ^{2
,
,}

1.
Faculty of Decorative Arts, Silpakorn University, Bangkok 10200, Thailand
2.
School of Economics, Shandong Women's University, Ji'nan 250300, Shandong, P. R. China

Received: 05 August 2025 Revised: 22 September 2025 Accepted: 25 September 2025 Published: 15 October 2025
MSC : 35Q91, 91G20

In recent years, numerous advanced technologies have been applied to financial research, yet few scholars have utilized Lie symmetry theory. Focusing on the fractal characteristics of financial markets, this paper constructed a fractal Black-Scholes (B-S) equation of option pricing incorporating He's fractal derivative, as well as studied its solution and dynamic properties through the Lie symmetry analysis method. First, the fractal equation was transformed into an equivalent continuous equation using the improved fractal two-scale transform. Then, the geometric vector fields, symmetry reductions, and exact solutions of the equation were derived. Finally, combined with the data from Linnan Dongsheng Film and Television Co., Ltd., the dynamic impacts of parameters such as fractal dimension, volatility, and risk-free interest rate on option prices were simulated and analyzed. The study found that the fractal dimension has a significant regulatory effect on the sensitivity of option prices, and volatility and risk-free interest rate showed phased influence characteristics by affecting the price fluctuation space and time value. This research provided a new theoretical method and analytical tool for option pricing in fractal market environments.
- fractal B-S equation of option pricing,
- Lie symmetry analysis method,
- improved fractal two-scale transform,
- exact solution,
- dynamic behavior
Citation: Yuan Yue, Chao Yue. Lie symmetry analysis for the fractal Black-Scholes equation of option pricing[J]. AIMS Mathematics, 2025, 10(10): 23346-23359. doi: 10.3934/math.20251036

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Abstract

In recent years, numerous advanced technologies have been applied to financial research, yet few scholars have utilized Lie symmetry theory. Focusing on the fractal characteristics of financial markets, this paper constructed a fractal Black-Scholes (B-S) equation of option pricing incorporating He's fractal derivative, as well as studied its solution and dynamic properties through the Lie symmetry analysis method. First, the fractal equation was transformed into an equivalent continuous equation using the improved fractal two-scale transform. Then, the geometric vector fields, symmetry reductions, and exact solutions of the equation were derived. Finally, combined with the data from Linnan Dongsheng Film and Television Co., Ltd., the dynamic impacts of parameters such as fractal dimension, volatility, and risk-free interest rate on option prices were simulated and analyzed. The study found that the fractal dimension has a significant regulatory effect on the sensitivity of option prices, and volatility and risk-free interest rate showed phased influence characteristics by affecting the price fluctuation space and time value. This research provided a new theoretical method and analytical tool for option pricing in fractal market environments.

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