Modeling complex financial dynamics via Lie symmetry and chaos analysis of the generalized Black–Scholes equation

Khizar Farooq; Ali. H. Tedjani; Muhammad Amin S. Murad; Yakup Yildirim; Taha Radwan; Khizar Farooq; Ali. H. Tedjani; Muhammad Amin S. Murad; Yakup Yildirim; Taha Radwan

doi:10.3934/math.2026343

AIMS Mathematics

2026, Volume 11, Issue 3: 8355-8381. doi: 10.3934/math.2026343

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Modeling complex financial dynamics via Lie symmetry and chaos analysis of the generalized Black–Scholes equation

1.
Centre for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia
3.
Department of Mathematics, College of Science, University of Duhok, Duhok 42001, Iraq
4.
Department of Computer Engineering, Biruni University, Istanbul–34010, Turkey
5.
Mathematics Research Center, Near East University, 99138 Nicosia, Cyprus
6.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia

Received: 09 February 2026 Revised: 15 March 2026 Accepted: 24 March 2026 Published: 30 March 2026
MSC : 37K40, 39B62, 33B10, 26A48, 26A51

The classical Black–Scholes equation has a central role in the theory of option pricing; however, the classical form lacks sufficient power to model complex non-linear market effects, such as sudden asset price movements, volatility clustering, and instability. In this work, we consider a generalized nonlinear Black–Scholes equation aimed at providing a more faithful description of these types of complex behaviors. Initial analytical tractability is obtained with the application of Lie symmetry analysis, which leads to symmetry generators and similarity reductions that reduce the partial differential equation under consideration to lower-dimensional ordinary differential equations. The resulting reduced equation is solved analytically using the generalized Arnous (GA) method, resulting in a spectrum of exact solutions, bright and dark solitons, and exponential and trigonometric families of solutions. The dynamical characteristics of the reduced system are then examined using bifurcation analysis and chaos diagnostics. Numerical investigations involve phase portraits, Poincaré sections, return maps, Lyapunov exponents, and bifurcation diagrams, which reveal transitions to different regimes (periodic, quasi-periodic, and chaotic) under the influence of external perturbations. These results prove that the generalized model can have complex nonlinear dynamics and multistability, which can provide information about the irregular fluctuations of option prices in financial markets. The contribution of this work is the coherent combination of Lie symmetry reduction, the GA analytical framework, and nonlinear dynamical analysis of the generalized Black–Scholes equation, providing an overall paradigm that could be used to explore complex financial dynamics.
- mathematical model,
- generalized Black–Scholes equation,
- Lie symmetry analysis,
- bifurcation analysis,
- generalized Arnous method,
- soliton solutions,
- chaotic behavior,
- multistability,
- chaotic attractor,
- Lyapunov exponent
Citation: Khizar Farooq, Ali. H. Tedjani, Muhammad Amin S. Murad, Yakup Yildirim, Taha Radwan. Modeling complex financial dynamics via Lie symmetry and chaos analysis of the generalized Black–Scholes equation[J]. AIMS Mathematics, 2026, 11(3): 8355-8381. doi: 10.3934/math.2026343

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Abstract

The classical Black–Scholes equation has a central role in the theory of option pricing; however, the classical form lacks sufficient power to model complex non-linear market effects, such as sudden asset price movements, volatility clustering, and instability. In this work, we consider a generalized nonlinear Black–Scholes equation aimed at providing a more faithful description of these types of complex behaviors. Initial analytical tractability is obtained with the application of Lie symmetry analysis, which leads to symmetry generators and similarity reductions that reduce the partial differential equation under consideration to lower-dimensional ordinary differential equations. The resulting reduced equation is solved analytically using the generalized Arnous (GA) method, resulting in a spectrum of exact solutions, bright and dark solitons, and exponential and trigonometric families of solutions. The dynamical characteristics of the reduced system are then examined using bifurcation analysis and chaos diagnostics. Numerical investigations involve phase portraits, Poincaré sections, return maps, Lyapunov exponents, and bifurcation diagrams, which reveal transitions to different regimes (periodic, quasi-periodic, and chaotic) under the influence of external perturbations. These results prove that the generalized model can have complex nonlinear dynamics and multistability, which can provide information about the irregular fluctuations of option prices in financial markets. The contribution of this work is the coherent combination of Lie symmetry reduction, the GA analytical framework, and nonlinear dynamical analysis of the generalized Black–Scholes equation, providing an overall paradigm that could be used to explore complex financial dynamics.

References

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