Introduction

In recent years, there are many researches use Lie symmetry analysis to partial differential equations (PDEs) which arising from physics, chemistry , economics and other fields [1–۷]. The investigation of exact analytical solutions of PDEs play an important role for a long time. Lie symmetry analysis is one of the most effective methods for finding the exact analytical solutions of differential equations, and many authors used this method to find the analytical solutions of PDEs [8– ۱۲]. Lie symmetry analysis method was originally developed in the 19th century by the Norwegian mathematician Sophus Lie and developed in differential equations since Bluman and Cole proposed similarity theory for differential equations in 1970s [3, 7]. Over the last forty years, there was a considerable development in PDEs which arise in mathematical finance [13–۱۹]. It was worth pointing out that Bordag and Chmakova’s pioneering paper studied the evaluation of an option hedge-cost under relaxation of the price-taking assumption by Lie symmetry analysis method. In particular, they found some analytical solutions to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity and showed that the typical solution would have a payoff which approximates a strangle, then used these solutions to test numerical schemes for solving a nonlinear Black-Scholes equation [16]. In this paper, we consider the following parabolic Monge-Ampere equation in ` the optimal investment theory [17]: usuyy + ryuyuyy − θu 2 y = 0, (1.1) where r, θ > 0 are constants. Existence of solutions to initial value problem for this equation were showed in [18]. The rest of the paper is organized as follows. In Section 2, we recall the model Eq.(1.1) arise from optimal investment of mathematical finance theory. In Section 3, vector fields and optimal system are given by employing Lie symmetry analysis method. In Section 4, the similarity variables and analytical solutions of Eq.(1.1) are obtained by using optimal system. Finally, conclusions are presented at the end of the paper.