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Abstract:We consider a family of equations with two free functional parameters containing the classical Black-Scholes model, Schönbucher-Wilmott model, Sircar-Papanicolaou equation for option pricing as partial cases. A five-dimensional group of equivalence transformations is calculated for that family. That group is applied to a search for specifications' parameters specifications corresponding to additional symmetries of the equation. Seven pairs of specifications are found.

A general nonlinear model of illiquid markets with feedback effects is considered. This equation with 2 free functional parameters contains as partial cases the classical Black-Scholes equation, Schönbucher-Wilmott equation, andSircar-Papanicolaou equation of option pricing. We obtain here the complete group classification of the equation, and for every parameters specification we obtain the principal Lie algebra and its optimal system of 1-dimensional subalgerbras. For every such subalgebra we calculate the invariant submodel and invariant solution, when it is possible. Thus, the series of invariant submodels and invariant solutions are derived for the considered nonlinear model.

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Abstract. We study the group structure of the Schönbucher-Wilmott equation with a free parameter, which models the pricing options. We find a five-dimensional group of equivalence transformations for this equation. By means of this group we find four-dimensional Lie algebras of the admitted operators of the equation in the cases of two cases of the free term and we find a three-dimensional Lie algebra for other nonequivalent specifications. For each algebra we find optimal systems of subalgebras and the corresponding invariant solutions or invariant submodels.