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Okay. As we said in the last video,

the Black-Scholes-Merton model deals with

the problem of pricing and hedging of financial options,

also called financial derivatives instruments.

This model is somewhat mathematically heavy and

amounts to solving some partial differential equations,

and it will take us too long to explain it this way.

Now, if you came to this course with a background

in financial engineering or mathematical finance,

you already know what the Black-Scholes model is.

But if you don't, remember that I said that no special financial knowledge is requested,

is required for this course.

In this case, I would suggest that you proceed in one of three possible ways.

The first way may be simplest one.

In this way, you'll just continue with this video then,

the next one and so on.

A simple model that I'm going to present in this lesson does

not require any knowledge of option pricing theory.

Yet, the second way is to watch this video to the end, and then,

take a look at the additional slides that I have for you for this week.

These slides explain the are binomial model,

a very simple and very popular discrete time

model that approximates the original Black-Scholes model.

These slides also give you some pointers

to where you can read more about the binomial model.

The binomial model is a very simple model that

explains the Black-Scholes model in terms of linear algebra.

Again, if you're already familiar with the binomial model,

it will help you to better understand what I will

be talking about next in more mathematical terms.

But if you're not, then what I'm going to present next is self-contained,

and does not assume that you know what the binomial model is.

So, it's really up to you.

And finally, the third way would be to stop the video now,

look first at the binomial model,

then read some more,

then come back here and continue.

So, let me wait a bit and give you some time to decide.

All right, I see you've decided to stay.

So, let's continue. I will try to explain

without any math what the Black-Scholes model does.

Now, how it does,

it will become clearer more later after I introduce a model

that is actually very similar to that binomial model that I mentioned to you earlier.

So, let's talk about the Black-Scholes model first.

So, we said in the previous videos that a simple model for

stock dynamics could be geometric Brownian motion with a drift.

Even though it may not be the best model for the stock price itself,

it captures the most important fact for the option pricing.

We said that the financial option also known as

a financial derivative is a financial contract that references

a particular stock whose payoff depends on what

happens with the stock at some future time T. For example, in one year from now.

But how do we know what should be

the fair price that you should be willing to pay for such option?

And here comes the main idea of Black-Scholes-Merton,

which is actually very easy to explain without any math.

They said the following: Let's assume that you

sold an option on 100 stocks of company ABC.

By these, you can meet to deliver 100 ABC stocks each for

price K at some future time T. What should you do?

Of course, you can buy 100 stocks now for

their today's price and just wait for time T to deliver those stocks.

But what if the ABC stock is too expensive now,

but you believe that the price will go down later,

closer to time T. Clearly,

buying those stocks now can be a very suboptimal strategy because it's very risky.

But if that's the case,

how about another strategy?

In this strategy, you proceed as follows.

First, you get some cash for selling the option.

With this cash, you buy only a few stocks and put the rest of cash in a bank account.

This is your replication portfolio for the options.

You can also view it as your investment portfolio,

which is made of just a few stocks of the ABC company,

and cash in the bank.

Now, once you created it,

you don't put money in it and you don't take any money out of it.

This is called a self-replicating or a self-financing portfolio.

All future operations with the stock investments will be financed

from this portfolio and without taking any money from it outside.

And vice versa, any gains from your stock trading will be put

back in the portfolio as additional cash in a bank account.

So, don't take any money yet out to get a new car from your stock trading.

Now, what do we need?

Why do we need the replicating portfolio?

You can use the replicating portfolio to track

how the option price which will

be the amount that you have to pay at time T changes with time.

The best way to understand why you need such tracking is to

think of the replicating portfolio as essentially an option position with the minus sign.

What does it mean? When you sold an option and

invested all your proceeds in the portfolio,

you did not create any wealth.

The value of your total portfolio,

which is the portfolio minus the option, is exactly zero.

It was now time zero when you just sold the option and bought the portfolio.

Now, imagine that you somehow are able to make sure that this will be always the case,

that is the value of your total portfolio is always zero no matter what the stock does.

This means that in this case,

the value of your portfolio of stocks and cash in the bank

will always be the same as the price of the option.

But then, it means that the price of the option today should also

be equal to the price of the portfolio today.

This is called in finance the Law of One Price.

If two securities pay the same in all possible future states of the world,

they should have the same price now.

So, to price an option,

we have to build a replicating portfolio such

that the total portfolio made of the replicating portfolio,

and a short position in this stock will always have

exactly zero value no matter what the stock does.

Now, we have only two questions left,

how to do it and what's the meaning of this?

Let's start with the meaning.

Why would you struggle in the first place to

create a total portfolio with exactly zero wealth,

and then bother to track it and trade stocks,

all just to ensure that you start with zero wealth and end up with zero wealth.

Such as zero wealth portfolio may sound like quantum physics

where particles and anti-particles are created from vacuum and then annihilate.

Yet, the answer given by the Black-Merton-Scholes model is

that an option price obtained in this way is an equilibrium price,

meaning that this argument gives

a fair option price such that no rational trader should trade below this price,

and no rational trader should trade above this price.

But by the way,

it's very interesting question why a rational trader should trade even

at the fair price of the options suggested by the Black-Scholes model.

But let's postpone this question until we explain how

the Black-Scholes model construct such a perfect replication portfolio.

Let's do it in the next video.