Studying a Tumor Growth Partial Differential Equation via the Black–Scholes Equation

Winter Sinkala,Tembinkosi F. Nkalashe

doi:10.3390/computation8020057

Abstract

Two equations are considered in this paper—the Black–Scholes equation and an equation that models the spatial dynamics of a brain tumor under some treatment regime. We shall call the latter equation the tumor equation. The Black–Scholes and tumor equations are partial differential equations that arise in very different contexts. The tumor equation is used to model propagation of brain tumor, while the Black–Scholes equation arises in financial mathematics as a model for the fair price of a European option and other related derivatives. We use Lie symmetry analysis to establish a mapping between them and hence deduce solutions of the tumor equation from solutions of the Black–Scholes equation.

Highlights

The study of the most common and malignant brain tumor, glioblastoma, known as glioblastoma multiforme (GBM), and that of option pricing, may be done in tandem
The connection through partial differential equations (PDEs) between mathematical models of glioblastomas and those of option prices can be exploited, courtesy of Lie symmetry analysis [10,11,12,13,14,15,16,17,18,19], to study one model through another arising from the “unrelated” field
It so happens that the Black–Scholes and tumor equations, both of which admit 6 + ∞ Lie point symmetries, can be reduced to the heat equation via an equivalence transformation

Summary

Introduction

The study of the most common and malignant brain tumor, glioblastoma, known as glioblastoma multiforme (GBM), and that of option pricing, may be done in tandem. In both cases, partial differential equations (PDEs) are the central vehicle for mathematically studying the dynamics of the phenomena. We use Lie symmetry analysis to construct a point transformation that maps the Black–Scholes equation to the brain tumor equation and every solution of the Black–Scholes equation to a corresponding solution of the brain tumor equation.

Preliminaries of Lie Symmetry Analysis

Concluding Remarks