Stochastic dynamical analysis, Monte-Carlo simulations, and waves dynamics of a coupled volatility and option pricing model under Brownian motion

This manuscript studies a coupled option pricing system called bidirectional quantum neural computation model, which is devised by Ivancevic to address both behavioural and efficient market dynamics as well as to incorporate the controlled stochastic volatility. The notion of stochastic process, i.e, Brownian motion is added to explore stochastic behaviour of the price option and volatility of the stock price. Some concepts of dynamical theory such as bifurcation, equilibrium points, chaos behaviour, and time series analysis are studied under the influence of the Brownian motion. Additionally, heat maps, Monte Carlo simulations (MCS), and stress testing for the considered model is illustrated graphically, indicating how stochastic system adapts to different market conditions. For waves analysis, we need to derive analytical solutions. Thus, we utilize sin-Gordan expansion (SGE) technique and Sardar sub-equation (SSE) method to attain some solitary waves of different families. With the help of these solutions, we describe the behaviour of the option pricing and volatility of the stock. The achieved results are simulated, via 3D and 2D graphs to show the impact of the noise term on the behaviour of solitary waves in financial market.