Now, for our case study, I'm going to assume the following parameters set. Spot of a 100, a strike of 80, specific is strike do not forget, you might say you can get various suffer instruction answer is yes, but I just want to focus on one strike at this point. Risk-free rate of five percent, dividend rate of one percent, maturity of one year. Then I'm having three examples. The first example is Black Merton Scholes. I don't want to go through the Stochastic differential equation. You may have seen it or not, I can simply go directly through the characteristic function, but I'm just writing it up. In a case of a Black Merton Scholes the assumption is that the stock price evolution follows the following Stochastic differential equations, what we call BMSSDE. Be sure, we're assuming that the follows, the following one. As you know, this is what we call drift part. This is what we call diffusion. This is the volatility of the process. This are interest rate, q dividend. Now, for this process, its characteristic function, that means the characteristic function of the log of the stock price is given according to this equation. Again, this is an imaginary number. That's the spot r q sigma maturity sigma t. Of course this new, this new that I'm having over here. Now in this example, I'm assuming that sigma is 30 percent. Now let me show you the table and I go through the code that they use to populate this table Then you have everything you can do exactly what I've done here. Let's see what I've done here. I discuss N, Eta, Alpha, and Beta. For Beta of this said, I would do it to be log of K, which turns out to be log of 80 because I'm using it for [inaudible]. This one is already gone. Then I have N, Eta, and Alpha. For Alpha I said Alpha should be greater than 0. Greater than 0 is strictly positive. Then I can go from something very, very tiny to something relatively big, thing. I'm going through this range and you can pick whatever you like. You can go smaller than this if you wish. You can go larger than this if you wish, I can leave it to you. As a matter of I can setup some, maybe it would be good to leave it as a quiz or something like that. Anyway, for Eta, I'm assuming two different choices. You can actually extend it if you wish, 0.1, 0.25 and for each of these, I assume one case and 2 to the power 6, one 2 to the power 10, 2 to the power 6, 2 to the power 10. That's exactly how I'm formulating it. I use these things and I populate this table. That means, for example, if you look at this one, that means for Alpha 1, N 2 to the power 6 and Eta 0.1. That should be clear. If you look at the population, some of them you've seen, I have it in yellow. Yellow means I already know that the true answer is this. How do I know it? Because for Black-Scholes, we have an analytical formula for it. Now, as you see when you look at this table, it seems that some cases I'm a bit off. If you miss actually, I'm doing extremely well here, why not? I'm going up to four digits. I don't have to, I could say up to a penny is good enough, but that should be fine. But as you see, for example, I'm pretty off here. I'm doing really, really badly, and I'm relatively off here too, but definitely Alpha two small is not a good thing. For Alpha two large depending on what N is. But it seems that this range, I've done really well. Actually I've done actually okay here as well. Then for Alpha between one to two, I'm doing well. That's as far as this. When it comes to eta, it depends on the choice of N. That means if the eta, for the case that for example, eta is 0.25, no matter I'm using N 2 to the power 6 or N to the power 10, I'm doing still well, the reason is because the cutoff is far enough. But if you are at the smaller eta, for N two small, I'm doing a poor job. Then maybe what you should do, I didn't do it here, for you is using eta 0.1 and N 2.8, which would become somewhere between to see how well you do. I leave it up to you because as you say you have the code, you could simply sit down and do these things and convince yourself. Let's look at the next example. I'm moving from Black-Scholes to something like Heston Stochastic Volatility Model. Heston Stochastic volatility model, I've having the SST the only reason I'm mentioning this because I had it for like Merton shoes. I want you to see the two STE, my apologies. The only thing I want you to notice is if I replace this by sigma, this look like exactly what Black-Scholes. Well, then it's not constant anymore. This itself follows another STE, if you don't know about Stochastic calculus or a stochastic processes, that should not be any problem whatsoever. The only thing I want to emphasize is that unlike Black moor tissues, the volatility is a stochastic that means it follows another process itself. That's what I want to emphasize and the only thing I'm saying here is this that, this is its characteristic function. The characteristics of log of the stock price is given by this, you may say, oh my god, this is very complex or conflict, that the answer is no, this is given to you. One more time see what we have. We have the S naught, r, q, T and you will see things sitting here or there. Of course, some cosine hyperbolic, sine hyperbolic. You don't have to worry about this. The only thing is given to you in close form, and of course I forgot to mention there is a gamma here, which that gamma is nothing but this, which is again given to you. For this example, this F5 parameter model, let's make sure that they know what those are. Is kappa, theta, lambda, that's it and there is a rho. The rho has to do with correlation between these two Brownian motions. That means we have two processes. One for the stock, one for the variance, and we wondering if two are working independently or these two are correlated. If they are independent, this would be zero, because correlation is between one and negative 1. In this case, I'm assuming correlation is 70 percent, minus 70 percent and the v naught. V naught has to do with initial point that you are starting your variance. That means that's initial variance. This is called the vol Laval and that says would be long-term reversion and that's to do with a speed of reversion. Anyway one thing you have to know is this, emphasizing one more time. My assumption is the characteristic function of the log of this stock price processes given to you and is a five-parameter model and for this parameter model, I'm assuming that's the parameter set. Now for this again, I will go and populate this table. Exact same table when it comes to alpha, eta and choices for N. Let's look at the population, I already know the true solution is this one, and is easy to see because it's nicely converges. One more time. It seemed that this range of series is good range, roughly speaking, the alpha range between one and two. But one thing you're noticing is, it seems that to get there, still I'm relatively doing well, I'm off by a penny, but not here though. One more time. If you remember, we had this in earlier as well, then eta is small. You have to make sure that you compensate on N. Maybe then you should try two to the power. I'm leaving this to you. You have the code. You can play with the code. You can convince yourself to see what the good choice would be. But it seems that this is a good range for alpha. Any thing between one and two should be good. Small is not good. When it's large, still you're doing okay. I mean, you're doing well, but then if you try 50 or 100, you will see actually that would then be a good choice. When it comes to ETA, seems that ETA 0.225 with two to the power ten for n, you would do a good job, which is consistent exactly with the previous case for Black Merton Scholes. Let me go through the last model is called variance gamma model. I'm not going to go through this history because it's a bit more complex. I'm having it over here. The only thing I want you to notice is what's interesting about model like variance gamma model is actually as opposed to diffusion. This model is what we call pure jump process. Without actually going through the formula I can tell you the difference between the two. Pure Jump process. A pure jump process or pure Jump processes are processes that the stock can actually jump or the stock prices jump, as opposed to diffusion processes that it is diffusing and you can actually have jump, but here's pure jump. That means you can have really, big jumps which would make it more realistic. These classes of models, variance Gamma or any variation of those processes actually give you a more realistic way of trying to capture the behavior of the stock market. For this process, the characteristic function is given by this formula one more time again, we have the S naught, r, q T, and the parameters of this process, which are sigma Nu and theta. I intentionally called the Nu here and I called this u for you not to get confused. If you remember, we always use some of these notations. I'm abusing Nu for the parameter of the model and using U now here, this u is dummy anyway, you can call it whatever you like. This is a three parameter model. The Sigma, Nu, and theta. Sigma again is volatility. Nu is what I would call kurtosis. Theta is what I call skewness. That means what the system says. It would allow you to have both kurtosis and skewness. The interesting thing is if you set these two equal to zero, you get Black Merton Scholes back. That simply means there is a built-in function to various gamma. If there is no jump in the market, which simply means I get rid of these two. You would go back to Black Merton Scholes, which makes it very realistic actually. At some point in next lectures in other series, I'm actually going to walk you through this and seeing that if you introduce some prices which coming from Black Merton Scholes, variance gamma can detect the fact that they're coming from Black Merton Scholes, that means if you calibrate it, you would get these two almost zero. The only non-zero would become the volatility. Anyway, said enough here, let's go to the table. That's the table exactly like before. Premiums for various values of alpha, n, and ETA. Again, exact same series for alpha, for ETA and for n as well. One more time is up to you. I did it 2-6 to 2-10 for n maybe I should have had two to the power eight. I leave it to you for you to do it. One more time. You've seen these strange, it's a good range. Actually can go even further, but they're going to be more consistent with other cases. Alpha between one and two. It seems that for ETA 2.5 and n being two to the power ten, you doing a very good job. I know this solution is this which I'm having it in yellow boxes. Good. Let me go through all findings and observations that you are having here.