Case, which is a case of a lognormal distribution. Lognormal distribution is a distribution for geometric Brownian motion, if you assume your process is geometric Brownian motion. If you don't know, that's fine, but I'm saying this is one of the simplest processes that we assume for evolution of a stock price process. Another name people use for it is actually Black-Merton-Scholes. Now, for the lognormal distribution, which again, conditional probability density function given the spot today, distribution of spot in future. Of course future depends on time T, which you will showing up here and there. And then I'm assuming for the stock there is a interest rate and dividend rate which I'm having it here, volatility which was showing up three different places, here and there. Now, the starting point, the final point, which will show up two places, and the rest is just another of exact definition of lognormal distribution. The interest is I can actually exploring what that one is. I'm assuming this is given to you, what you ask is price, the option for it, call and the put. Now, before we move to the next step and showing you the results, it's very important to recognize, depending on what today's spot is, you may see a different shape. For example, I'm having four different shapes here for you. Now, this one, the blue one is the case that the spot today is 50 as you see is jammed around the 50, or at least you will see a big pick. It gets flattened out as the spot has started moving to the right. For example, for the case, this is the flattest one, this red one you see is for the case that the spot today is 200. As you see, is much flatter than the one that is 50. It's very, very important because if you're starting at 50, for you to get 200 probability is much, much lower than you start from 200 and you want to get to say, 220 or 230. That's why it's much flatter. Good, now for this case, for the problem I'm going to have in mind, I'm assuming today's spot is 100, risk free rate is 5%. I'm assuming dividend rate is 1%, and of course you can play with it yourself in this environment. Today, people assume typically interest rate of being about two and half percent or something like that, or just assuming 5%. Maturity of one year, assuming T is 1. Again, you can change it to anything you like. Volatility 30% or .03, and strike of 140. Again, you can change it to whatever numbers you like, this is just for the example I'm having. Now, for the call, as I said, for the call I'm picking in an eta independently. Then what I'm doing here is, I'm having various different values of n, and I'm showing all as 2 to the power. I don't have to do it this way, the reason is because that's my habit. Specifically, when we go into other methods you will see. An eta, I'm assuming from just .25., which is 25 cents going to a dollar. That means when it comes to breaking it down into subintervals, that would be your step size. And then definitely you can build your KB, which would be K + N eta. I don't have to emphasize on this one in the case of a call, you remember we mentioned this earlier. Now for this one, I'm simply showing what you would get from that numerical integration or the so called approximation. And these are the values that I'm having actually. I'm going through all of it, the only thing is using some of them I'm putting in the yellow. That means that the exact answer is actually, if I'm not mistaken, is this. How do I know it? Because for this case we have the analytical one as you see. For most cases we've done extremely well, except we're off here and we're off here, which simply means you need to make sure that because maybe here B wasn't big enough, large enough. Because if you substitute, and maybe B wasn't large enough, but the rest of them you see is large enough and the step size is good enough that you get a very good accurate approximation. Then as I said, that was the call prices for strike of 140, which is out of the money call for various values of eta and N. And overall, we've done a good job. Let's go to the case for the put. For the put, as I said, we are just picking or choosing N and we would have implied eta which was simply doing K divided by N. Do not forget, that's why when I'm having various different N, I simply getting eta, which would be the 140 divided, say 2 to the power 8. If you do that, that's what you would get, and there is as well. Now, the premium I'm getting, actually these are the premium. And as you see for all actually we're doing a very good job there. All I'm putting it in a yellow one. I mean, they're actually pretty, pretty tight for the in the money case. Because when we have in the money, out of the money call for the same instruct, the put become in the money. Now, before we end the lecture and we go through the Python code, let's do an assessment of this approach. The very first question is, how good is or how good was this approximation? For appropriate choices of N and eta, the approximation is pretty good. Actually it was pretty tight. How feasible is this approach? That's the second question. The thing is this, if you have the conditional probability density function in closed form, you have it analytically. And it's not that expensive to calculate it because in some cases you might have it, but it becomes expensive to calculate it. It's absolutely feasible, it's actually feasible. But in most cases, it turns out that it might not be available in an integrated form or in close form, then overall this approach is not feasible. I just wanted to say, if you have it, definitely you have a way to do it. If you do not have it, you should think about an alternative way. The next question immediately after that is, is there an alternative or a better approach? The answer is yes. That's exactly what I'm having in mind to go over for the next 4 lectures, which will be starting transform techniques. Then what I'm going to do is, I'm going to go over the Python code. And what I did in those two tables, the population and all of it that I did, I'm going to actually go through the code and explain to you how to do it.