Abstract This paper reveals new types of exact solitons to the nonlinear class of generalized Black–Scholes equation. This model has much importance in the field of financial mathematics and financial engineering. Black–Scholes economic model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. By utilizing the modified simplest equation method, we investigate the distinct kinds of exact wave solutions involving, trigonometric, hyperbolic, and rational functions. To obtain and verify the solutions, we use the Mathematica software. We demonstrate the obtained solutions through two-dimensional, three-dimensional, contour, density graphs by using the Mathematica tool. The main contribution of this paper is to analyze the qualitative analysis, including modulation instability, bifurcation analysis, chaotic behavior, and sensitivity nature. We explore the important characters of chaos through 2-D phase portrait, 3-D phase portrait, Poincare sections, time series, and Lyapunov exponent. At the end, the gained solutions are fruitful in different areas related to finance, like economics, financial engineering mathematics, financial engineering and many others.
Raheel et al. (2026) studied this question.