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We know that we can get a closed form for European option price. And we can calculate directly the normal distribution accumulation. But I saw that people use many approximation methods such as Fourier transform ... What are the reasons? Please explain to me in detail, I do not have an full understanding but can obtain interpretation from basic ideas.

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We can only get closed-form solutions under certain assumptions about the market dynamics, e.g. in the Black-Scholes framework (share prices follows a GBM), the European option can be valued with the well-known Black-Scholes formula.

For other assumptions where no closed-form solution exists or is known (e.g. share price is a Levy process), FFT methods are used to arrive at approximate solutions.

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In short, FFT methods are used for efficiency and need for generalization to more complex dynamics and products. Reference below should help, but there must be many more publicly available.

https://www.ricam.oeaw.ac.at/specsem/sef/events/program/slides/oosterlee_cm_sm.pdf

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