Analytic solution of a nonlinear Black-Scholes equation with price slippage

Abstract

We study a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of transaction cost and a price slippage impact that lead to market illiquidity with feedback effects. After reducing the equation into a second-order nonlinear partial differential equation, we find that the assumption of a traveling wave profile to the second-order equation reduces it further to ordinary differential equations. Solutions to all these transformed equations facilitate an analytic solution to the nonlinear Black-Scholes equation. We finally show that the option is always more volatile compared to the stock when 1∓√1−(1− )2 (1− )2 < S0 S ert.