Abstract. We analytically present the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black-Scholes model. These rogue wave solutions may be used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

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The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.The nonlinear Schrödinger (NLS) equation is a foundational model describing numerous nonlinear physical phenomenon in the field of nonlinear science such as optical solitons in optical fibres [1,2,3], solitons in the mean-field theory of Bose-Einstein condensates [4,5,6], and the rogue waves (RWs) (also known as freak waves, monster waves, killer waves, giant waves or extreme waves) in the nonlinear oceanography [7,8,9,10]. RWs are single waves generated in the ocean with amplitudes much higher than the average wave crests around them [11,12]. Recently, the RWs have attracted more and more attention from the point views of both theoretical analysis [21,22,23,24,25,28,29,34] and experimental realization [13,14,30,31,32,33]. The oceanic RWs can be, under the nonlinear theories of ocean waves, modelled by the dimensionless NLS equation [7,8] i ∂ψ ∂t + 1 2which describes the two-dimensional quasi-periodic deep-water trains in the lowest order in wave steepness and spectral width. In addition, it has been shown that the RWs can be generated in nonlinear optical systems and the term "optical rogue waves" was coined by observing optical pulse propagation in the generalized NLS equaiton [13,14]. Some types of exact solutions of Eq. (1) have been presented to describe the possible formation mechanisms for the RW phenomenon such as the algebraic breathers (Peregrine solitons) [15], the time periodic