Study of the risk-adjusted pricing methodology model with methods of geometrical analysis

Abstract

Families of exact solutions are found to a nonlinear modification of the Black–Scholes equation. This risk-adjusted pricing methodology (RAPM) model incorporates both transaction costs and 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 RAPM model. Reductions are given in the form of different second-order ordinary differential 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.

Keywords:

• transaction costs
• invariant reductions
• exact solutions
• singular perturbation

AMS Subject Classification::

• 35K55
• 34A05
• 22E60

Acknowledgements

The author is grateful to Michael Nechaev and Alexandr Yanovski for helpful discussions.