The present model describes a perfect hedging strategy for a large trader. In this case the hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using Lie group theory we reduce in special cases the partial differential equation to some ordinary differential equations. The Lie group found for the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described. © 2008 Springer-Verlag Berlin Heidelberg.