# solution on the time domain becomes "periodic" after the inverse fourier transform

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.

• What do you mean by periodic? It doesn't look like it repeats to me ... Maybe try plotting the logarithm of the $x$-coordinate on the $x$-axis, and then it might be sinusoidal? Nov 12, 2019 at 4:37
• @ZubinMukerjee 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. Nov 12, 2019 at 4:45
• This is not an artifact of where you stop the plot? Nov 12, 2019 at 4:59
• @ZubinMukerjee No, that’s the full time domain Nov 12, 2019 at 5:08
• maybe you should have a look at this question math.stackexchange.com/questions/2432207/… Nov 14, 2019 at 16:43