In d'Halluin et al. (2005) (http://imajna.oxfordjournals.org/content/25/1/87.short) the authors claim that the correlation integral$$ I(x) = \int_{-\infty}^{\infty} V(x+y) f(y) dy $$can be approximated with FFT methods by$$ I_{k} = IFFT((FFT(V)_{k})(FFT(f))_{k}^{*}) $$where $(.)^{*}$ denotes the complex conjugate.I have tried to program this in Matlab with $V = \max(90 e^{y} - K,0)$ and $f$ being the probability density function of the normal distribution with $\mu = 0.1$ and $\sigma = 0.2$. Analytically I get the result $I(0) = 8.783$ but in whatever way I try the FFT approximation the results are nowhere close to that. My math background is really weak and I don't have any intuition or knowledge regarding the FFT method. I calculated a set of values (powers of 2) for $V()$ and $f()$, did both element-wise and matrix multiplications of the FFTs of the two vectors and checked the real part of the IFFT, but I never get the desired result.What am I doing wrong? Can someone tell me how I get to the result?Here's the code of one of the ways I did it:S = 90;K = 100;           mu = 0.1;sig = 0.2;AA = 2.5;N = 255;             x = (linspace(-AA,AA,N+1))';V = max(S.*exp(x) - K,0);ffff = normpdf(x,mu,sig);I_k = real(ifft(fft(V).*conj(fft(ffff))));
Approximating improper integral with FFT
matlab;fourier transform;integration