Well, let's now proceed with our generalizations. First of all, we talk about replication again, and this approach led to the famous formula that is known by the names of the people who first derived it as Black and Scholes formula. It was derived in the early 70's and then later the people who contributed to that were awarded the Nobel Prize. Unfortunately, by that time Fischer Black had already died. Therefore, Myron Scholes and then Robert Merton, who contributed to that later than at the point when the formula was first derived. These people were awarded the Nobel Prize. Now, the idea was very fruitful. The idea was the creation of a dynamic portfolio of a stock and a bond. Because we saw in the previous example in the previous episode that when we have all of the upper and lower outcome that is not realistic. In reality, there is a whole distribution of these outcomes. Again, we know that in finance, people try as hard as they can to use the normal distribution because this is the one that can be described with very few parameters. However, when you deal with the optimum pricing you have to keep in mind that what is actually distributed normally is not the stock price but rather the change of the stock price because the stock price is limited by zero. The prices of stocks cannot be negative. However, the percentage changes on this prices can go as low as minus infinity. Therefore, what they did, they did first of all, they created a dynamic portfolio of the stock and bond and then, they used the lognormal probability distribution. Again, this is the one, then like I said, the percentage changes are distributed normally. Well actually, the equation that they derived through that that was a very advanced equation that is in partial derivatives and that uses the real stochastic weiner process. But like I said, this is something that a lot of people read, a lot of people talk about, but few people understand. Well, we will not go deeper in that but we will say that the result was this famous Black and Scholes formula, that I'll produce on the next charts. But before I ever tell you that, even with this huge leap frogging insight in what goes on, what Black and Scholes we're able to do, they produced a formula for validation of European option, for the stock with no dividends, and with volatility sigma and risk-free rate as constants. So will see after we produced this formula that this very case is very important. But it's not the most interesting for a lot of people who deal with optional investing and option valuation. So the formula looks like this. Again, it's kind of intimidating. Again, I will reproduce here. So Black and Scholes works for European option with sigma and R as constants and no dividends. Then, it looks like this. On this list I will show it in not the most intimidating way. I will say that the price of a call option, a European call option with all of these components. Constant looks like this. This is N of d1, S, when the S is a share of stock, minus N of d2 times PV of K, where K is the strike price. Now, what are these mysterious ends, the d1, d2. By what formula or by what approach we calculate the PV of K? Well first of all, N(x), this is a so-called probabiility integral so basically, it looks like this. This is 1 minus square root of 2 pi integral minus infinity to x, E to minus Z squared over 2 dz. So this is a special function. But it can be easily illustrated. So that looks like this. If this is a normal distribution, so this is minus infinity, this is plus infinity and this is x. So this end of x, this is the area, under this curve until the point x. So this is a very widely used in probability theory. Now, you can see that this is end of x and now here we have d1 and d2. Now, so far, before I flip over I can say that end of d1 is the equivalent of Delta. So this is the proportion of the stock in this dynamic portfolio. And this is the proportion of the load and this is the negative sign here. Well unfortunately, it becomes, as Alice would say, curiouser and curiouser. So d1 is equal to the natural logarithm of S over PV(k) divided by sigma square root of t and then plus sigma square root of t over 2. And then, d2 is basically the same as just d1, minus sigmas square root of t. Now, we calculate PV using continuous compounding. So PV(K) is equal to ke to minus rf times t. This is basically it. You can see that there's a lot of math, and some special functions, and all that only to be able to come up with a formula for a European option with a constant sigma and RF and with no dividends. Well, so far so good. But how do people use this? Well clearly, option traders, they are not mathematicians. But these formulas were put in financial calculators because you basically have to just calculate some of these contributing parameters and then with the use of computers or even calculators, in the early 70's, you can easily come up with the corresponding values. So all that is done and used widely. Now, the next question is how do people cope with the idea that this is not an American option and most traded options are American? Well, there have been specifical approaches developed that allow to some what refine this formula. Because the equation, if it's refined, then it cannot be solved numerically. So then what the people did, they just used the solution for this European option and then somewhat refined how that can be a little bit changed. The result of that can be changed for the American option and for some changes in volatility, for example. But that's basically, I showed that only for information because basically all these things, they use their formulas and tables and people can easily use that in reality. But now let me say a few words about generalization of this formula because we see that it's not enough to use this Black and Scholes because there are certain areas in which Black and Scholes clearly fails. We said that it's, let's say, drawbacks. It's always easy to talk about some problems when you see that some people did a great job for the drawbacks of Black and Scholes. Well first of all, like we said, it is no good for American options. Now, it's no good. For stocks with dividends. And we can say that maybe it's going to be not of that much value for bond options either. Because, for example, for bonds options, if you assume that the risk-free rate is constant for some period of time, then the changes in bond prices, they are based on changes in the risk-free rates. So that seems to be. So, I'll put it right away that is, no good for fixed income options. Well, we can say that, so, it's always easier to criticize and say that for which this is not that much good, and that leads us to another thing. We can say, well, let's say a few words about another approach that leads to the sort of famous binomial method. And binomial method looks like this. It is a generalization of our risk neutrality approach. So, remember, we started with point zero and then we said that we can arrive at point one, and we had like two outcomes here. Well, in reality, you can divide this period. Let's say, like this. And then, you can say that if for example, there you divided into four pieces, then, you can arrive at the four-way outcomes. So, let's see. This is the upper and lower way. Right? From here, you can go here or you can go there. From here again, you can go here. So you can see that basically, you can arrive at one, two, three, four, five outcomes. And it is going to be some distribution here. Like this. And what if we keep dividing? Now, you can see that as each node here, you can calculate the value of an option at this node. Now the next question is, how can we arrive at this node? What's the probability and how it can be dealt with. And again, remember, I told you that these numbers, like 122 and 82, and I said, these are mysterious numbers. Well, there's not that much mystery about that, but it can be shown that if you use log normal distribution that people use here, then for each and every node, then, you have the following formula, 31 plus rH is equal to e to sigma square root of h small. And then, I'll say just what it is. And then, one plus r low is e to minus sigma square root of h. Well, h small here is basically the number of these intervals or actual the inverse of that. So if there are two intervals, then h will be one half. If they are 100 intervals, it will 100. So, now you can see that if you take one plus ariation, then one plus RL, then the multiple of these numbers is going to be always one. So that's why 82 and 122, if you multiply one by the other, you will have 100. So it's basically, so if this number is this, then this number will be 100 over this. That is why, like I said, if I put that 120, then this number will not be 80, it will be somewhat higher. For that reason, I use these numbers. Now, in reality, what people do, they say that here at the point of exploration, they say, we know at all nodes, we know the value of an option. And then, we'll start to go back step by step. Now, for stock option, it's one procedure for bond option it's even worse, because you have to really say so, you start somewhere with some interest rate. And then, from this interest rate, you make some conclusions, and then you go always back and from these numbers because you discount at each and every period. And then, you may arrive at the right thing or may arrive at the wrong thing. And if you arrive at the wrong thing, then you leave all this. So the approach that is called a binomial approach, as you can see, in terms of formulas is not as bad as Black and Scholes, but it's extremely cumbersome. And actually, you can find that in many textbooks, or let's say, especially for fixed income. And you can see that formulas are just ratios. But then, with the number of periods going up, it becomes extremely cumbersome, and people use all the numerical approaches, and that is how they proceed. However, you can see that if we study that at each and every node, we can easily generalize that for exercising, because, for example, if here, we can see that it's better to exercise and get a large enough dividend, we will go ahead and do so. And then, we will replace the number at the node by the cash flow from this dividend, instead of just having what the value would be if properly discounted. So you can see from this discussion that, unfortunately, this is not so easy to implement in reality, but we're not pursuing this goal in this course. It's much more important for all of you to understand where to go and from where to get these approaches and numbers. What is important you can see how the option value is being contributed by the parameters and how it actually adds up. So here, I will stop with the idea of option valuation, and starting from the next episode, I will show you how this very idea is used in order to add some value to the real investment projects in which we have this nice opportunity to change some of our decisions in the future or to employ several of the newly arrived information.
