Well, let's analyze the following situation. You have the ability to build a plant that produces some high technology equipment, computer or whatever. And you have a forecast of cash flows. You analyzed all that. Let's say, this is not a very long-term project because this very computer and this very technology may soon become obsolete. So you are planning for a couple of years, let's say for three years. And then you ended up with the unfortunate, finding that the NPV of this project, let's say, at 20 percent. 20 percent is the corresponding rate of discounting, which is okay for this kind of project is negative $40 million. We know the decision. Don't do this. However, let's say that on top of this project, we have another project that we can engage in three years. So what are the terms of this project? Well, I could have produced some cash flows but, instead, I will put that all in the wrapped up way. So, first of all, decision to invest in this is made in three years. Now, let's say investment in this project was $450 million, but now, we have investment in new projects are $900 million. And the PV of these investments, at point three, PV at three is $800 million. So, basically, the story is like this. We sit at zero, this is one, two, three here. You have the ability to invest 900, and then the PV of that at this point is 800. So, clearly, this is a bad project because it has an NPV. Well, but also we are in a really risky volatile industry, and we say that the volatility of cash flows is 35 percent. Now, without talking about option valuation, one of the arguments behind the idea of even discussing this, because if you recall that the NPV is negative 40 million, is the following. If we do not engage in this project with the negative NPV, we can stay out of this industry. We can lose connection with that technology development, and, therefore, in three years, if we are not in this project, we will not have the option to invest in any project of that kind. So we are sort of out of this segment of the industry. But these are sort of strategic considerations. But let's take a look at that, and on the next list that I'll just show you how we can identify all the situation as an option and what will be the value drivers of these options. Well, we say that this is actually the call option, and what are its value drivers? Well, first of all, t, which is easy, is three years. Now, k is also easy. It's $900 million. Well, we output here to clear things. Sigma is 35 percent, which is great. And then let's say that rF is 10 percent. Remember that r, for the risky investment, was 20 but that rF is 10 percent. Now, it's a little bit trickier with S because S is the stock price and we have to see it right now. Well, what we do is this is PV now at 20 percent of the PV that we will receive at 0.3 So this is basically 800 divided by 1.2 to third power, which is $463 million. So this is the stock price. Now, what we know that in order to be able to apply the optional valuation formula, we also have to find the PV of K. Well, the PV of K goes at the risk-free rate. So this is 900 divided by 1.1 to third power, which is 677. And then we can calculate some of the ratios. Let's say S over PV of K will be equal to 0.68. This is just the preparation to use the Black and Scholes formula. And then C over S is point 0.119. And, finally, I'll put it in the web that the value of such a call option is approximately $55 million if we use Black and Scholes. Now, we see what happened. Remember that the value of our project was negative $40 million. So, now, we can say that our actual project is this negative 40 million, plus this option which is 55. So, on top of that, altogether we have positive $50 million. So this very option, because of its length, because of high volatility of cash flows, this is very important, and because of these parameters, this is an out of the money option. But still because see that S is 463, but the strike price is 900, so this is a deep out of the money option. However, it has a significant value here. So what we did, we sort of identify that this opportunity to make the future investment is actually a call option. We have these parameters, and when we use Black and Scholes to evaluate that. Well, you can always say that, the use of Black and Scholes maybe is not perfectly justified here, but we made some assumptions. And this is sort of, because, for example, if we found this number to be, whatever, 40.1 million, then clearly the validity of our statement that this together is a positive and the project will be much lower. But, here, you have a significant change, and it's a qualitative change. Before, it was a negative of the project. And, now, it's a positive of the project. So that gives you an idea how sometimes people can use some very simple approaches and very straightforward calculations to change this story completely and to see that, actually, this project has a positive NPV. Well, in reality, you may always deal with some more complex situations, and very often, you will have to use a binomial approach to study some of the outcomes, and that will give you a much better result, although requires some more hard work. But I will sort of wrap up here because I have shown that the option theory and the option valuation approach really allows you to sometimes change your decisions or, at least, find two new decisions, and then go ahead with projects facing uncertainty. We have some better background if you will. So that's about how this works. And in the remaining episodes of this week, I will say a few words about how the option theory of the option valuation approach is used in the area of fixed income securities, namely of the risky debt. We talked about that for government securities, for riskless debt before in the second week of this course. And, now, and then I promise that, we will come back to that, and now I'm trying to deliver my promise. And in the next two episodes, I will, in that sort of descriptive way, show the realm of this analysis, and I will also show some huge challenges of that.