Analytic Solution of a Nonlinear Black-Scholes Partial Differential Equation

Abstract

The assumptions under which the standard Black-Scholes equation has been derived are restrictive (e.g. liquid and frictionless markets). When illiquidity and market friction are introduced into the market, financial models based on these assumptions fail. Nonlinear equations for modelling illiquid markets have been solved numerically. Numerical techniques give approximate solutions. Recently, Lie group symmetry analysis has been used to solve the same. Although Lie group symmetry analysis is very useful in determining all the solutions of a given nonlinear equation, it has been established that any small perturbation of an equation disturbs the group admitted by it. This in effect reduces the practical use of symmetry group analysis. Our objective is to find an analytic solution of a nonlinear Black-Scholes equation for modelling illiquid markets. The methodology involved transformation of the nonlinear Black- Scholes equation into a groundwater equation. This yields Ordinary Differential Equations which have been solved. Using substitutions and integration led to an analytic solution of the nonlinear Black- Scholes equation. In a real market situation, this solution may help in finding how typical prices of derivatives can be described hence contributing significantly to the field of Financial Mathematics.