Analytic solution of a nonlinear Black-Scholes partial differential equation

Abstract

We study a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of a feedback effect. This is an illiquid mar- ket effect arising from transaction costs. An analytic solution to the nonlinear Black-Scholes equation via a solitary wave solution is currently unknown. After transforming the equation into a parabolic nonlinear porous medium equation, we find that the assumption of a traveling wave profile to the later equation re- duces it to ordinary differential equations. This together with the use of localiz- ing boundary conditions facilitate a twice continuously differentiable nontrivial analytic solution by integrating directly.