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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?

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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 $S_T$.

For European style options, Fourier methods are extremely popular; in particular because they apply to a very wide range of stochastic processes (much more than just Black-Scholes, Merton and Heston). In particular exponential Levy processes have a simple characteristic function. Some models introduce however new difficulties (e.g. mutlivalued complex valued functions (Heston trap) or residue calculus).

Another downside is that it first requires one to study Fourier transforms and this may take too much time in a lecture. On a first glance, it also occurs less economically intuitive than tree methods or MC simulations. So, it may be just for pedagogical reasons. After all, Fourier methods are still 'younger' than numerical PDE approaches, simulation and trees. For different asset classes, simulation and trees are popular (e.g. interest rate models whereas Fourier methods are less used there). So, it may also depend on the focus of your lecturer.

Finally, some models simply do not have a characteristic function available, e.g. local volatility models. So, you cannot really apply Fourier methods here. For rough volatility models, one has to approximate the characteristic function.

However, there’s some deep financial interpretation and there are some extensions to apply Fourier methods to Asians and other path dependent options. So, they are a popular tool in 'advanced' option pricing.

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