The study of phase portraits, multistability visualization, Lyapunov exponents and chaos identification of coupled nonlinear volatility and option pricing model

Abstract

In this study, the coupled nonlinear volatility and option pricing model is examined. A leverage effect is produced, indicating a negative correlation between stock returns and volatility, and a confined Brownian motion linked to the nonlinear Schr & ouml;dinger equation is exhibited. This model is considered a coupled nonlinear wave substitute for the Black-Scholes option pricing model. A mathematical strategy is introduced to comprehend market price fluctuations for the suggested model. Consequently, the necessary parameters for the existence of these solutions are revealed. The obtained numerical results of market price are discussed through graphs to illustrate and validate the theoretical findings. The generalized mapping approach of Riccati equations is applied to the model under consideration. Several periodic and singular soliton solutions are successfully constructed for the model. When appropriate parameters are chosen, both 2- and 3-dimensional plots that graphically represent some of the observed waveform solutions are included. Additionally, bifurcation, chaotic analysis, Lyapunov exponents, and multi-stability are performed to gain deeper insights into the related dynamical system. Phase portraits of market price fluctuations are shown for various parametric values of the corresponding dynamical system and at the equilibrium points. The results demonstrate that slight changes in initial conditions lead to price fluctuations in the model.