# Choice of grid for numerical integration

I have to compute an integral involving the characteristic function for pricing options in a model and it so happens that accurate approximation seems to be mostly about putting lots of points in between 0 and 1, and it almost doesn't matter what you do further down with the point grid.

Since we're talking about 1000s of such integrals, I am wondering if there is some systematic way of doing something like that so that I don't waste time on points that don't help.

• Are you asking if there's a "good enough" number of intervals to calculate or asking how big a range is "good enough", meaning if the function is defined for all numbers but 99.9% of the area is between 0 and 1? Oct 19, 2021 at 14:37
• You could go either down the Fast (fractional) Fourier Transform route (or related methods), effectively discretizing your integral and using fast FFT implementations, or if you truly want to integrate numerically, you could go for some adaptive integration scheme that can handle oscillatory integrands, e.g. Gauss Kronrod. Effectively, the adaptive integration scheme "selects" the relevant points for you. Oct 19, 2021 at 18:06
• @Stanley I am asking about whether I could find some scheme for selecting points for numerical integration such that loads of them are close to zero, but it becomes very sparse further down. Oct 20, 2021 at 16:19
• @Kermittfrog I'd have to check those out to see how it would work in my case, but those suggestions are certainly welcome. Oct 20, 2021 at 16:23
• Without additional information on the (location of the highly) oscillatory behavior of the integrated, I’d suggest an adaptive scheme tbh. It is not too computationally expensive, and you get an error bound on your integral, too. Oct 20, 2021 at 17:12