Group classification of a generalized Black–Scholes–Merton equation

•Group classification of a generalization of the Black–Scholes equation is performed.•We obtain and use the equivalence transformations to simplify the classification.•We obtain invariant solutions for the barrier option problem for two nonlinear cases.

The complete group classification of a generalization of the Black–Scholes–Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each nonlinear case obtained through this classification, invariant solutions are given. To that end, two boundary conditions of financial interest are considered, the terminal and the barrier option conditions.