solution on the time domain becomes “periodic” after the inverse fourier transform

Question

I was trying to solve european option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf).

The final step of this algorithm is to transform the solution on the frequency domain to the time domain by inverse fast Fourier transform, which means that I can retrieve approximations for all the grid points.

However, I noticed that only the solutions in the center of the time domain (in this case, x=40, and I used log grid points) are accurate. Here is the plot: the orange line represents the approximation using FFT, the blue line represents the real solution.

Moreover, the approximation looks periodic. So I was wondering what kind of situation will inverse FFT give a 'periodic-like' solution? I did some research on this topic, and foundthat DFT assumes the input is periodic, which may cause "time domain aliasing" (https://www.dspguide.com/ch10/3.htm).

I also found that "spectral leakage" may be related to my problem. Can anyone give me some advice on this? Thank you very much.

P.S. By periodic, I mean the solutions on the two ends are very close to each other. I changed lots of parameters, and this always holds.

Answer 1

Multiplying a kernel and signal spectrum in Fourier domain lead to a circular convolution and not a linear convolution, so in order to it become linear convolution you must zero pad your signal and kernel before taking the Fourier transform (up to M+N-1 where M is the signal's length and N is the kernels's length). (if you compare the blue and orange signals, the beginning of orange signal has higher values because of of higher value at the end of blue signal and it has lower values in the end because of lower values in the beginning of the blue signal).