Analytic solution of a nonlinear Black-Scholes equation

Abstract

We study a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of transaction costs that lead to market illiq- uidity. After reducing the equation into a nonlinear parabolic porous medium type equation, we find that the assumption of a traveling wave profile to the porous medium type equation reduces it further to ordinary differential equa- tions. Solutions to all these transformed equations together with the use of localizing boundary conditions facilitate a twice continuously differentiable so- lution to the nonlinear Black-Scholes equation. We also find that the option is always more volatile compared to the stock. All the risk parameters except Gamma are negative throughout time t.