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

TL;DR

This paper applies Lie group analysis to a nonlinear Black-Scholes modification, deriving exact solutions and reductions that incorporate transaction costs and portfolio risk into option pricing.

Contribution

It introduces a novel geometric analysis approach to solve and reduce a nonlinear risk-adjusted pricing model with exact solutions.

Findings

Derived families of exact solutions to the nonlinear model

Identified the Lie algebra and symmetry reductions of the RAPM equation

Provided explicit and parametric solutions for various reduced equations

Abstract

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.