Explicit solution of Black–Scholes option pricing mathematical models with an impulsive payoff function

This paper deals with the construction of explicit solutions of the Black–Scholes equation with a weak payoff function. By using the Mellin transform of a class of weak functions a candidate integral formula for the solution is first obtained and then it is proved that it is a rigorous solution of the problem. Well known solutions of option pricing value problems are obtained as particular cases of the solution proposed here.