Document Type : Research Paper

Authors

• Usaamah Obaidullah
• Sameerah Jamal

University of the Witwatersrand, School of Mathematics, Johannesburg, South Africa

Abstract

This paper studies the nonlinear quantum-probability based Schrodinger type, Ivancevic options pricing model using the method of Lie symmetries to determine its point symmetries, invariant analytical solutions and conversation laws. In our analysis, we consider a non-zero and zero adaptive market potential model. We demonstrate that this model is invariant under a five-dimensional Lie algebra for the former, and invariant under a seven-dimensional Lie algebra for the latter case. These symmetries allow for a progressive reduction of the equation and thus facilitate a solution. We obtain reductions, exact solutions and conservation laws for both the non-zero and zero adaptive market potential models. We show that many exact solutions are expressible in terms of two transcendental functions, the Fresnel sine and cosine integrals. Graphical solutions are provided in certain cases. This analysis and solutions to such a financial derivatives pricing model are unique, providing novel insights.

Keywords

• Lie symmetries
• Exact solutions
• Schrodinger equation
• Conservation Laws