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

In this paper, we investigate the non-linear Black-Scholes equation: ut+ax2uxx+bx3uxx2+c(xux−u)=0,a,b>0,c≥0.and show that the one can be reduced to the equation ut+(uxx+ux)2=0by 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.