Linearly Autonomous Symmetries of a Fractional Guéant–Pu Model

We study the group structure of the Guéant–Pu equation of the fractional order with respect to the price of the underlying asset variable. It is one of the models of the dynamics of options pricing, taking into account transaction costs. The search for continuous groups of linearly autonomous equivalence transformations is carried out. The equivalence transformations found are used in constructing a group classification (within the framework of linearly autonomous transformations) of the equation under consideration with a nonlinear function in the right side of the equation as a free element. In the case of a nonzero risk-free rate, it is shown that two cases of Lie algebras of the equation under study are possible: two-dimensional in the case of a special type of free element and one-dimensional in the remaining cases. If the risk-free rate is zero, there are four variants of the Lie algebra, which can be two-, three-, or four-dimensional. In the future, we assume to use the obtained group classification in calculating invariant solutions and conservation laws of the model under study.