In this paper, we investigate the non-linear Black--Scholes equation: u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0. and show that  one can be reduced to the equation u_t+(u_{xx}+u_x)^2=0 by an appropriate point transformation of variables. For the last equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of this equation. Using the obtained results , we get a number of exact solutions of the Black--Scholes equation  .
In this paper, we investigate the non-linear Black--Scholes equation: u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0. and show that the one can be reduced to the equation u_t+(u_{xx}+u_x)^2=0 by an appropriate point transformation of variables. For the resulting equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of the equation. Using the results obtained , we get a number of exact solutions of the Black--Scholes equation under study and apply the ones to resolving several boundary value problems with appropriate from the economic point of view terminal and boundary conditions .
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