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.