In Carr-Madans option pricing method, why do they use FFT?

Question

In the famous fourier option pricing method by Carr-Madan, (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf), the crucial formula is

They evaluate this by using the trapezoid rule to write it in a form to which FFT can be applied. The output of the FFT will give us the call prices for multiple strikes. The complexity is $O(N \log N)$.

But the problem is that we have little control over the strikes. In particular if $N$ is large, the strikes become spread out, and many of the values are completely useless to us. Really, we are only interested in strikes close to $S_0$, the asset price.

So would it not be better to just calculate the integral above manually using quadrature rules, and then repeating that process for any given strike? This would be a $O(NM)$ algorithm with $M$ being the number of strikes, but in practice it would be very fast since we can re-use a lot of the computations (notice that $k = \log K$ is independent of $\psi(v)$ in the integral).

Answer 1

Indeed, the FFT was a notable improvement in computational option pricing in 1999, but further investigation has shown that it can be easily optimized both in terms of speed and accuracy.

For instance, this paper compares the traditional FFT with a strike optimized version of the Carr-Madan formula (CM-OPT), concluding that the CM-OPT is simultaneously faster and more accurate that the FFT.

Finally, the comparison between the FFT and the CM-OPT deserves a special mention. While both are based on the same pricing approach, the CM-OPT's flexibility allows (i) pricing any required strikes, (ii) choosing any integration size and technique and (iii) avoiding interpolation biases. As a result, the CM-OPT is both faster and more accurate than the FFT, thus rendering this method inefficient. Based on our results, we see no reason to employ the FFT over the CM-OPT, but further analysis may be needed in order to confirm this hypothesis.

Source: Crisóstomo, R (2018): Speed and biases of Fourier-based pricing choices: a numerical analysis, International Journal of Computer Mathematics, 95:8, 1565-1582, DOI: 10.1080/00207160.2017.1322691