Qualitative Dynamics and Exact Traveling-Wave Solutions of the Nonlinear Black–Scholes Equation

Abstract

The extended nonlinear Black-Scholes equation is a significant modification of the conventional option-pricing model that incorporates nonlinear market effects caused by transaction costs, liquidity limitations, and feedback mechanisms. In this paper, we give a complete qualitative and analytical analysis of this model utilizing nonlinear dynamical systems theory. A traveling-wave transformation converts the governing partial differential equation into an autonomous planar system with an equilibrium structure, local stability, and Hamiltonian formulation determined by an appropriate conservative condition. The resulting first integral allows for a comprehensive phase-space analysis and the derivation of accurate traveling-wave solutions. Specifically, explicit Jacobi elliptic solutions are found, while the classical solitary wave is recovered as the limiting homoclinic orbit, indicating that localized and periodic waves belong to the same family of Hamiltonian solutions. Global phase portraits are created to demonstrate the qualitative influence of solution trajectories on model parameters and to offer a geometric interpretation of analytical solutions. The periodically driven system is statistically explored using Poincaré sections and bifurcation diagrams, demonstrating the breakdown of invariant tori and the emergence of Hamiltonian chaos as the forcing amplitude grows. The proposed Hamiltonian framework combines equilibrium analysis, accurate traveling-wave solutions, global phase pictures, and chaotic dynamics into a single theoretical environment. This methodology offers a systematic way to researching nonlinear option-pricing models and is easily applicable to other nonlinear evolution equations in applied mathematics and mathematical finance.