# When do Fourier inversion methods run into problems?

So in my courses, we always priced options either with Monte Carlo methods, or some sort of PDE discretization.

Then I looked up Fourier inversion methods on my own that rely on the characteristic function, and they're shockingly effective (see Carr-Madan 2000).

European option prices are obtained in milliseconds, and very accurately, with exponential convergence, and the methods are extremely simple. No need to worry about setting up a Euler-scheme for simulation, no need to worry about Rannacher time-stepping in Crank-Nicholson PDE algorithm .... just implement the characteristic function and calculate a simple sum.

So ... what am I missing? What are the drawbacks of these Fourier methods? Why would anybody use anything else for European option pricing?

When do Fourier methods fail?

Pricing path dependent options (Asians, lookbacks, barriers, Americans) is much harder with Fourier. MC simulations are easier in these cases. Recall the characteristic function only contains information about the terminal stock price $$S_T$$.