Welcome to option pricing, bio transform techniques, lecture 7 by now you know why I call this transform techniques because what we are doing is we using Fourier transform for doing this. In this lecture, which is the last lecture. But I'm doing is now I'm going to go through options pricing by a 50 word, explain everything, but literally what I'm doing is I'm going through the implementation part and then I'm going to price for three different processes. I'm going to price the options, option premiums I would say for different strokes, different maturities. But as specifically for different parameters, because if you remember we discuss about ETA, we discuss about N capital, and which was 2 to the power N and Alpha. But you wonder how sensitive the options prices would be with respect to various different choices of Beta, N, and Alpha. That's what I'm actually trying to do here in this lecture. After I went through all the slides, what I would do is I would walk you through the Python code for this lecture and explain to you how I actually, I did that now then, let's do a recap of what we've done so far. What I would say is having the characteristic function of logarithmic of stock price process. I keep saying this because my assumption is this is given to you. After I went through all the slides, what I would do is I would walk you through the Python code for this lecture and explain to you how I actually, I did that now then, let's do a recap of what we've done so far. I'm assuming this is like a black box, exact like an engine. You have the engine you just sitting down to drive the car. You don't have to be a mechanic to get to the engine. One can price a European is very important to mention this, one can price European be not talking about American options are the options that you would have the right to exercise at maturity. No time in between. That's the difference between European and American. I want to make sure that we're clear here via FFT. And when I'm saying this not only for one is struck. That means for fixed maturity. As I said, you can do it for various different strikes. That's the benefit of actually using FFT. Now again, I'm borrowing this from previous lecture, lecture 6, having characteristic function we choose a tub. want to make sure that emphasize we choose N, and beta that means the user who's ever using it, the trader, the risk management who is using it. They have to choose ETA, N, and beta. Of course you can come up with some kind of learning algorithm that would do it for you and you will see as we go through. The slice is easy to build these things, but that this phone I'm assuming the user is actually picking ETA, N, and beta and then as soon as you do this then you calculate Lambda from the constraint. You remember this and the constraint was, this constraint that means once this and this gets picked a to N you have to calculate Lambda from that constraint you setting new up. You also set Alpha. You form that vector. You remember this from lecture six. I don't have to go through it. This is again I borrowed from the previous lecture. You pass it through a 50. Getting wiped out by now. You comfortable with FFT. Not only I call it on the library, I did the explicit implementation. You have the code, you exactly know how that works, and once you get the y out, you just get the real part of it for each component or each entry. My apologies, you multiply by the corresponding damping here, which for of course the first one is K1, K2, and K3 and you remember. Case and I'm going to discuss beta soon. And that would give you a vector of N call options for those strikes. But maturity, don't forget maturities fix at this point. Now what will be the choice of beta? Going back again, beta is the very, very first strike. And then when we set the set this one up. We go through this and each through discrete points. Each of these is Lambda. That means this would be your K1, K2, K3 to the last one which is KN. And it turns out the K1, the very first strike, would be exactly beta. That's how we design it. There are two common choices. Now I'm going to have this one here. I'm going to discuss it, but I want to make sure that for the quiz I'm going to set it up for you to make sure that you can convince yourself that I'm right. Now if you want middle of a range. Corresponds to the at the money. That simply means the point that K= S note. Then the way you would set up beta as the choice would be. Log off is note, which is a spot price minus N over 2 Lambda. That means you know that you have N of these. This is the N, the N over 2 middle of the range comes over here. You want the middle range to coincides at the money, that means you want the cap hub Sally K to be equal to S note. Don't forget I'm doing their larger space because everything is at log striker space. That is why that means this is actually what I'm doing here. Log of K in the middle to be equal to the spot. This is something that I want you to leave it as an exercise for you. Convince yourself that's the case. That's one case. The other case is, if you want the very very first call coming out of this, that means when you're looking at this vector. CTK1 to CTKN the very first one you want to correspond to a specific strike a you say beta to be logged that K. A fixed K that you have in mind. Then these are the common choices, and you will see the code as I'm going through it. I'm going with the second choice, I mean there is no reason, that's what they decide to do it
