All right, let's now proceed with have some examples of option valuation.

We will put that as follows.

First of all, this is replication and we will try to value the following option.

Let's say that the stock price now is $100,

and we are valuing at call option with the strike price key of $112,

and we will also assume that the risk free rate is eight percent.

Now, that's not enough,

this information is not enough to come up with this valuation.

So, what I will also assume is the following situation.

So, now we sit with

the $100 stock price and I will say that there are two outcomes in the future.

The upper way, when we have $122,

and then on the lower way, we have $82.

You can say why are these numbers like this and why are they not let's say, 120 and 80.

Well, I will show just a little bit later

that there is some relationship between these numbers,

so we cannot put them very arbitrarily.

But let's now compare the following portfolios.

One portfolio will be,

so, I will use one of the two approaches.

One will be, I will buy a share of stock but I will also take a loan,

and the loan will be enough to pay for the stock in the lower outcome.

So, I take the loan,

that is the PV of this 82,

which is how much?

This is 82 divided by 108 because this is the risk free rate, which is 75.93.

This special loan.

And then, comparing two portfolios.

One is the stock price minus this PV of this loan.

I'll put it like this.

And the other part is the call option.

Let's see how much these portfolios are worth in both cases.

First of all, let's start with this in the upper case.

So, here we have 122

minus 82 is the amount that we repay the loan because we borrowed this.

But then, let's say in a year,

this is zero point.

This is point one, whatever. At point one, we have to pay back exactly 82.

So you subtract 82,

that gives you $40.

Now, in the lower case,

you have 82, then you subtract 82 and you have zero.

So, these are the values of this blue portfolio in this case.

Now, let's take a look at the call option.

Now, here you have,

122 is the stock price,

and then you have to subtract the strike price minus 112,

that gives you $10.

If however, the stock arrives at 82 that is lower than K,

so here, you have zero.

Look at these two portfolios.

This one gives 40 and zero and that's the one that gives 10 and zero.

So, you can say that the blue portfolio is equivalent to four red portfolios.

Now, in finance, if two different portfolios give

you the same cash flow at one period in time,

then you can say that they are equivalent.

Now from that, we can conclude that, let's say,

I'll put it 4C is the

same as S minus PV of this loan.

Now, this is at point one.

Now, let's see what happens at point zero.

At point zero, we can rewrite it and say that C is equal to

one quarter of S minus PV of this loan.

And now, let's see at point zero,

S is 100 and PV of loan is 75.93.

So, that gives you 6.02 for the price of the option.

So, we created a portfolio that replicates the option.

And then we said that if it does, then clearly,

we know how much this portfolio is worth at point one,

which is 40 and zero.

And therefore, it should be equivalent at point zero.

And at point zero,

we know how much this portfolio is worth and that is the way how we evaluate the option.

Now, there are some observations to that.

Well, number one is, by the way,

this one quarter has a special name and it is called Delta.

This is basically what percentage of the stock you have

to buy in order to replicate this call option.

It plays an important role in option evaluation,

option strategies but we just have to have an idea about that.

Now, some questions arrive.

Question number one is why these strange numbers?

Again, I promise to you I will answer this at some future point in time.

Question two, how we can generalize this because clearly,

there are just two outcomes here and what if there are many more than these two outcomes?

Well, this is sort

of a stable Delta but what if this Delta changes over time?

Now, these generalizations with some of these ideas,

and if there are many more outcomes,

all that leads to the derivation of the famous Black and Scholes formula,

that I will talk about just a little bit later.

But for now, I will show to you how we can reach

the same result with the use of risk neutrality approach.

I'll use the blue,

so this is risk neutrality.

Again, we have all the same inputs.

But now, we say that,

this is 100, this is 122, this is 82.

By the way, in the previous approach,

I said nothing about the probabilities of

this upper or lower outcomes. We did not need that.

We dealt only with the replication and discounting at the risk free rate.

Now, here let's say that Pi is the probability of the upper outcome,

and we say that the result of that is that the stock price goes up by 22%.

And then clearly, one minus Pi will be the probability of a lower pay off and here,

we have minus 18%.

Now, we can say that we can calculate what is the probability times the outcome,

and then the average expected return for the investor.

We can say that on the upper path with probability PI,

there is plus 22%.

On the lower path with probability one minus Pi,

this is minus 18%.

And again, if we assume risk probability,

it all should be equal to eight percent,

but be reminded that this is the risk free rate.

Now, if you solve this equation with respect to Pi,

you will get that Pi is equal to .65,

and one minus Pi will equal to .35.

Now, let's take a look at how we can value the option here.

We know that that the lower path at point one,

the value of an option is zero.

But on the upper path,

the value of an option is $10.

Remember, that's the same from this.

So here, C is equal to $10.

Now, let's say, we just calculated by discounting back.

We can say that the call option value at zero is equal to,

this is a probability of upper path Pi

times the value of the option at point one which is 10,

and then divided by 108.

And if you calculate it,

.65 times eight, divided by 108,

we will get exactly the same 602.

Well, clearly, there is no magic here

because the approach dealt with the same numbers and therefore,

we are not very much surprised.

However, the approaches are very much different,

and how can we generalize here?

Well, clearly again, there is going to be some distribution of outcomes,

and then another thing is that if this is a period,

you can divide it in some shorter periods and you can arrive

at this point and some other points here in between over different paths.

This arrival to these paths that leads to the more generalized binomial approach,

that as we will see in the next episode,

is actually extremely fruitful and allows to deal

with very complex situations that are in really,

very much of interest for investors.

And in the next episode,

equipped with these simple examples,

will generalize them and talk about the Black and Scholes approach and binomial approach.

And these two will lead us to the,

not only to the understanding of how this works but how

people come up with some advanced evaluations.