Symmetry reduction and exact solutions of the non-linear Black--Scholes equation

In this paper, we investigate the non-linear Black–Scholes equation: $$u_t+a{x}^2{u}_{xx}+b{x}^3{u}_{xx}^2+c(x{u}_x-u)=0,a,b>0,c\ge 0.$$ 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.