Families of exact solutions are found for a nonlinear modification of the Black-Scholes 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 correspondingly the set of invariant solutions to the model. On this way we can describe  complete set of possible reductions of the nonlinear RAPM model. Reductions are given in  form of different second order ordinary differential equations. In some cases they can be reduced as well to first oder equations . We discuss the property of these reductions and  corresponding invariant solutions.
Families of exact solutions are found to a nonlinear modification of the Black-Scholes 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 correspondingly the set of invariant solutions to the model. In this way we can describe the complete set of possible reductions of the nonlinear RAPM model. Reductions are given in the form of different second order ordinary differential equations. In all cases we provide solutions to these equations in an exact or parametric form . We discuss the properties of these reductions and the corresponding invariant solutions.
R
for
to
R
On
In
A
the
A
the
R
some cases they can be reduced as well to first oder equations
all cases we provide solutions to these equations in an exact or parametric form
R
property
properties
A
the