Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis

Families of exact solutions are found to a nonlinear modification of the BlackScholes equation. This risk-adjusted pricing methodology model (RAPM) incorporates both transaction costs and the risk from a volatile portfolio. Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM equation. It gives us the possibility to describe an optimal system of subalgebras and the corresponding set of invariant solutions to the model. In this way we can describe the complete set of possible reductions of the nonlinear RAPMmodel. Reductions are given in the form of di↵erent second order ordinary di↵erential equations. In all cases we provide exact solutions to these equations in an explicit or parametric form. Each of these solutions contains a reasonable set of parameters which allows one to approximate a wide class of boundary conditions. We discuss the properties of these reductions and the corresponding invariant solutions.