In this paper, the conformal fractional derivative and the Stratonovich integral are used to investigate the fractional stochastic Bogoyavlenskii equation with multiplicative noise. Using the dynamic system method, the chaotic behavior, sensitivity, and precise traveling wave solutions are provided. This study presents a complete investigation of the periodic, quasi-periodic, chaotic, and sensitive behavior using perturbations analysis, phase trajectory analysis, time series, and Poinca'{r}e section. A number of novel traveling wave solutions are developed, including Jacobian elliptic functions, exponential and hyperbolic functions. The effect of multiplicative noise and fractional derivatives on the solution is then examined using a comparative analysis technique. The results show that the fractional derivative and random noise have a considerable impact on how the model behaves. The methods and results in this paper are useful and can be applied to studies of even more complicated phenomena.