Dynamics of Lie symmetry, Paul-Painlevé approach, bifurcation aalysis to the Ivancevic option pricing model via a optimal system of Lie subalgebra

The Ivancevic model offers more accurate option pricing by including more grounded assumptions about the behavior of the market. Distinct forms of exact solutions are produced through the use of the analytical Paul-Painlevé method for the $(1+1)$-dimensional Ivancevic option pricing model. A five-dimensional Lie algebra is produced, with scaling and dilation as the remaining point symmetries in space and time. We employ symmetry reduction of Lie subalgebras to derive closed-form invariant solutions. In certain reduction cases, we convert the chosen model into a spectrum of non-linear ordinary differential equations (ODEs), which have the advantage of providing a large number of closed-form solitary wave solutions. Moreover, bifurcation theory is used to analyze planar dynamical systems at distinct equilibrium points, observing deviations under external perturbations and sensitivity under various initial conditions. Through these analyses, financial models can be made more robust by understanding how changes in model parameters or market conditions impact the dynamics and stability of option prices. These methods collectively enhance the understanding, robustness, and accuracy of option pricing models, providing valuable tools for both theoretical research and practical applications in nonlinear dynamics.