Introduction to Volatility, Correlation and Copula - YouTube

As usual, you can download the files to today’s discussion by simply browsing this link

So just simply go to this link

and proceed to click on ‘volatility, correlation and copula’ to get the files

You can double click on it

(double click on the little information tab right here)

These are the topics that we will be discussing today

Other than the script used on this website, you can perhaps use the one on Google Collab (if you are more familiar with that)

Double click it, and it should look something like this

If you are having trouble with opening the link, you can try reopening the link that I’ve mentioned earlier and clicking the ‘volatility, correlation and copula’ menu

Here is the outline of all the topics that we will be discussing today

We'll talk about the definition of volatility,

and if we’re referring to volatility, what is our exact purpose? What are we trying to see here?

Actually, we can simply estimate volatility by using what we call as ‘standard deviation’

Aside from that, there are other ways to modelling,

like Exponentially Weighted Moving Average or EWMA for short,

GARCH (Generalized Auto Regressive Conditional Heteroscedasticity)

Volatility can also be implied from fx option prices (option transaction)

Have you heard of the term ‘option’ before?

Well, we can buy a currency option like buy Dollar call Rupiah put for example

A type of volatility that we obtain from the value of the option is called an ‘implied volatility’

Volatility is not constant; and we need to be aware of this

So if we’re talking about option, especially the volatility against the strike price,

it will not be constant like this. On the contrary, it would look like this

This curve right here is called ‘volatility smile’

Which means that the volatility for every strike most likely not the same

because the volatility right here is higher than the rest

There’s a term we call as At The Money strike volatility which is normally it is the volatility at its lowest point

Another ones are ‘out of the money’ and ‘deep in the money’ volatility which may be higher and would form a volatility smile

That is the reason why volatility surface exists; it shows us how to look things in a 3-dimensional chart,

and that a volatility would be different in accordance to its strike and period of time

Moving on, we will be discussing about correlation and copula

We use correlation to see the dependency of one variable to another (the relation; the linear relation)

Although, it’s best to keep in mind that a zero correlation does not always bring independency

because what we will be seeing here is this correlation right here

Spearman correlation is only referring to its linear relationship

But if we’re talking about a relation in the form of (for instance) the square of x or x to the power of three,

the correlation might be equal to zero

However, we do know for a fact that it has a certain relationship, therefore it would not be independent

And for that reason, we must differentiate independency from correlation

Though, if it really is independent, then the correlation would be equal to zero

But do keep in mind that correlation applies only if we have two random variables (where each of them are normally distributed)

For example, if we have v1 that is normally distributed and v2 that is also normally distributed,

we can then obtain joint distribution

We can actually get it straight away by using a formula

Now the problem is, what if v1 and v2 are not normal?

How can we find the join distribution?

On cases like this, we must need the function called a ‘copula’

There are a lot of variety of copulas; like gaussian, student-t, etc

But there is a variety called multivariate gaussian copula; and we use it when we have more than two or three variables (v1, v2 and v3)

So these are all random variables that are not normally distributed

The types of distributions include triangular distribution or lognormal distribution, etc

Now, how do we obtain the joint distribution?

How can we use a joint distribution?

If we referring 'credit risk’ for example, we must first obtain the distribution of credit loss

In credit risk, it is certainly the credit loss distribution is not normal (not normally distributed)

For that reason, one-factor gaussian copula exists and is widely used and even used in Basel II as well

where minimum capital requirements under IRB Basel II uses one-factor gaussian copula

That will be it for today’s discussion! (:

Any questions?