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.
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.
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.