Some notes on the solutions of BSM equation

The solution of Black-Scholes-Merton (BSM) Partial Differential Equation represents the model for pricing an option. It is a very useful application for trading terminal. The solution gives the theoretical value of an option (Call/Put). In the present paper we apply Fourier Transform Method to solve the equation for plain vanilla payoff function and Log payoff function, which are the boundary conditions for the BSM partial differential equation. Also, we observe and show that averages of these two payoff functions will give exactly the average of two solutions. And we also extend this result.