Let me just put everything together, when it comes to implementation, start from scratch. What we said is this, we said having characteristic function of log of this stock price process. One more time, this characteristic function is given to you, it's provided and what we are building here is actually model-free setup, that means once you have that characteristic function, you don't have to do anything, you just pass it through and everything would work exactly like a black box. We choose eta, that's what's clear. The eta was the sampling size that we did. We come up with N, which is typically we write as two to the power small n. We also choose beta. As I said beta has to do with you is your choice, which I'm going to go over that. We calculate lambda, because once you set up eta and N, we have this constraint. This constraint was needed to link our calculation to FFT algorithm. Once you set this one up and that one up, then you get the implied lambda, which simply by solving it for lambda which is two pi over N eta. That's exactly what I am having here. We're setting the step sizes over space of Nu, which if you remember, was Nu J was J minus one eta. We said also alpha, you remember that damping factor just so-called for some alpha greater than zero, I'm going to come back to you and tell you that not any alpha would work, you'll see it soon. What we do is this, we form a vector. That vector is this x_j that we had. If you remember, let me just go back to this slide. This is what I call x_j. I set this one up, come over here, and the way I setting it up is, if you go back to this, that's exactly what, how each element of that vector would look like, except for the first one, which is eta divided by two, the rest is just eta. There is exactly the same, except is moving with the index, that means for X_1 is Nu one, Nu one, Nu one, Nu one. For X_2 I have Nu two, Nu two, Nu two and Nu two and the same would go to Nu N, and do not forget that the Nu was J minus one eta, that means the Nu one, for example, would become just simply zero, because when you substitute one Nu two would become just eta Nu three would become two eta, and it goes on and on. This is, as you see, is simply a model free setup. The reason is once you have this characteristic function you evaluated that these arguments and you just forming this vector. Once you form this vector, the next step is this. You pass that vector through FFT routine. One more time, the FFT routine is you simply using the libraries in Python, which is FFT. You calling FFT, you passing this X vector which was X_1 to X_N. The output is a vector Y, which would look like Y_1 to Y_N. What you doing is knowing that options prices would then be imaginary numbers, you pick the real part of them for each components, you multiply it by these, which goes with k_1, k_2 to k_N, and that would give you the options prices for various different instructs one more time, don't forget that the K_m was nothing but beta plus M minus one lambda. You remember them because lambda from that constraint, which was two pi over N eta and beta is you picking beta, which in the next lecture, I will explain to you how to find beta. What I'm going to do is this. I am going to refer you go to Lecture seven, the last lecture on the series, I'm just going to actually go through two Python quotes for you, which I'm explaining how to call FFT, but I'm going to do it two ways. One I'm going to do it by simply writing it out. The one is by writing it that means, an explicit implementation versus calling the Python FFT and just show it to you that we would get exact same results. That means you can do the explicit implementation that nobody would actually do it. Everybody is typically calling it because that would become actually the efficiency of grabbing this FFT algorithm from the library, either in Python or any language that you're using. Thank you.