Now, in this final episode of the fifth week, I will complete the description, a very short and brief description of the approaches to valuing risky debt. We will talk about ideas about valuing bond options. So let me put it here. Valuing bond options. As I have previously mentioned, unfortunately, this is an advanced area that definitely goes beyond the scope of this course. But I would like you to have the idea of how people tackle this problem in general, and there are numerous areas in which you can get some more information about that. First of all, again, let me reiterate that in most cases for bonds, Black and Scholes doesn't make any sense. In some cases, it does, in some special cases, but most often people use binomial approaches that are more relevant to bond option valuations. And now we deal with a random Rs, and let's take a look at the case of a callable bond. Well, we will take the view of an investor. And you know that an option to call the bond back is a valuable thing for the issuer. So for the bond holder, it's sort of a bad thing. So we can say that the callable bond is a non-callable bond minus an option. And there may be some formulas that I will show for information. So formula one, that the price of a callable bond is equal to the price of a non-callable bond, minus the price of an option. Well, this is sort of clear. Now, the price of a non-callable bond oftentimes is if not easy but straightforward to find. The price of an option is a much more difficult thing. Now, we are trying to sort of approach this price with the use of our good old friends, duration and convexity. And now we can say that the duration of non-callable bond is equal to duration of a callable bond, I am sorry, like this, is the price of a non-callable bond, divided by price of the callable bond, now, one minus delta duration of a non-callable bond where delta is the delta of this option. And it's even worse for convexity. I will just give you the idea without going deeper than that, that the convexity of a callable bond is equal to the price of a non-callable bond, divided by price of the callable bond, times here comes convexity of a non-callable bond, one minus delta, and here, minus price of a non-callable bond gamma, and then there's gamma, and then this duration of a non-callable bond squared. Well, I specifically, without put that much emphasis on that, the gamma is another parameter of an option. So you can see that formulas become progressively bad, but they, unfortunately, do not help us much because the question stays, how can we approach this option valuation? And like I said, in general, we unfortunately have to deal with these binomial models. And here I will just put the overall chart that gives you the idea. If this is yield and this is the price and this is the price at which the bond can be called. So the non-callable bond is something like this, but with the callable bond, you can see that here, they became sort of similar, but then it goes like this. So this is the value of an option. Well, and on top of that, on the market we can observe some of these prices and from these prices we can extract things like option adjusted yield. So, this is the yield for the bond compared to the yield of a non-callable bond, then also there's option adjusted spread. This is the difference between the yield of a bond with an option and bond without an option, then option adjusted duration. We can do that too. We can do even option adjusted convexity. All these things, they can be found but again the procedures by which they are found, they are cumbersome. There these parameters are important for the people who study fixed income securities and they all are based on various on the approach to option affiliation. So, that must be kept in mind. Again, I'm not trying to explain something to the final point, it's unrealistic unfortunate. I'm just trying to broaden the horizon to give you an idea that all this is dealt with, by the people who study them. Now, a couple of special words about Mortgages and CMOs. CMOs as you can remember stands for Collateralized Mortgage Obligation and we talked about that in the first course of this specialization. But for now, like I said the most important thing is that we hear deal with prepayment and that can be a rational. A Rational this means that I prepay when the rates go down, because for example I took out the mortgage at 5 percent and now the rates drop to 3 percent. I can borrow the amount of the remaining principal at 3 percent and then repay this whole balance to the original mortgage originally or to whoever is now the recipient of my payments and then I will be paying much less. Well, this is good for me as the mortgage taker. This is bad for the people who manage mortgages, for whom mortgages are assets. Now, and then like I said there is also the QFDR, Quit Fire Dire Retire. This is, these are the cases in which mortgages are prepaid not because of the change of the interest rate. Now, and when we talk about the CMO, we know that the CMO is a derivative instrument in which they are a cash flows of some of the pool of mortgages that are used as collateral for the holders of these cash flow. We are dealing with super risky instruments and not only that, there are strictly speaking here, few knowns and many unknowns. And therefore, because basically what you have to do in order to approach CMO valuation you have to study all underlying have to study all the prepayment rates, you have to create the forecast of how these underlying mortgages are prepaid and then from that, the second step would be to see how these cash flows that are derivative from the cash flows of collateral they reach the holders of CMO branches. Here you are dealing with most of the valuation models there. You basically make some forecasts and you calculate the expected cash flows along these paths and then you take the weighted average along many paths. This is really a modeling case and something that goes well beyond the scope of simple valuation. Basically, you see here there are quite a few unknowns but that's unfortunately not it. Now, and let me finally say some words about bond options. So far, I just showed that there are quite a few options that arise there and like I said we, in the valuation we use binomial methods or I would put approximately binomial. Sometimes these methods even the more advanced, but the good news about binomial methods is you know that it allows at all nodes to openly take into account the fact that some options may be actually exercised. And that is important here, because for example sometimes if these bonds contain options, let's say, the option to put or the option to call then you can see that if at a node you can see that it becomes beneficial for the investor to exercise the option or the otherwise, it's beneficial for the issue to exercise the option then you replace the price at this node with the exercise price. Now, we said that special case is CMO options and here we said that, there are options in underlying and then ways to follow cash flow received by final holders, final trans-holders. And the final thing, that I would put here again because unfortunately, this is the endless area I would say that so far we talked about some options that were valued using some of these approaches. But then, there's also many path dependent options. I'll give you an idea for example there sometimes there may be options that, let's say they lose value if the price of the underlying goes beyond a certain threshold or goes below another threshold. If it stays between, then it still preserves its value. So basically, if there is one path from point 1 to point 2 which is, within this range then the options still exist there. But, if there is another path that hits either the upper or lower boundary then at this point the option disappears. So, there is a huge class of option valuation models that deal with building these paths and calculating the value of an option along this path and then using corresponding distributions to come up with a certain triple quota weighted average if you will. As you can see from this final discussion in this week the complexity and the real challenges of fixed income instruments valuations when these instruments contain options. This is really endless. What I'd like to say by that is just to open your understanding a little bit to the, to this realm that this is really huge area and this area is quickly developing because investment banks and the rest of the companies they keep introducing new and more complex instruments and almost all of them they contain very special options. This area is the one in which many interesting things can be actually implemented. Now, I'm wrapping up this week at this point and in the next the final week of this course, we will come up with certain final conclusions, we will take a really brief look at what we've done so far and I will send you some positive messages about as I had promised before, how you can benefit from the knowledge of corporate finance in your life and careers. There are some assignments with respect to this week some of them are quite challenging with option valuation. So, I wish you good luck with them and I'll see you all next week.