Symmetry analysis of a model for the exercise of a barrier option

A barrier option takes into account the possibility of an unacceptable change in the price of the underlying stock. Such a change could carry considerable financial loss. We examine one model based upon the Black–Scholes–Merton Equation and determine the functional forms of the barrier function and rebate function which are consistent with a solution of the underlying evolution partial differential equation using the Lie Theory of Extended Groups. The solution is consistent with the possibility of no rebate and the barrier function is very similar to one adopted on an heuristic basis.