Zero sample correlation does not necessarily mean there is no linear relationship - page 19

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Avals:


The logarithm of price increments seems to be clear, but the logarithm of price is not clear either

Do you count price increments as absolute or relative? The logarithm of relative increments equals the difference in logarithms of prices. This is the reason why the price itself is logarithmic.
 
Mathemat:
Logarithms are used to explicitly establish that some quantity with a distribution resembling a normal distribution has a lower bound of zero. In deriving the Black-Scholes formula, it is assumed that the price distribution is lognormal, i.e. it is not the price that is normally distributed, but its logarithm.


This does not mean that it is necessarily logarithmic. I could be wrong, but I think BlackScholes is options https://ru.wikipedia.org/wiki/Модель_Блэка_-_Шоулза

Any transformation must have a meaning (a purpose) to reveal something, to find something that is not visible in the original set of numbers.

 

hrenfx, have you tried building the scatter plot of those two rows after which you decided to create this thread? ;)

 
Prival: That doesn't mean you have to logarithm it. I could be wrong but I think Black_Sholes is options https://ru.wikipedia.org/wiki/Модель_Блэка_-_Шоулза

I have seen the output of this formula. It relies precisely on a lognormal distribution of the price of the underlying option asset. There, among the underlying assumptions, is the assumption that the price of the underlying is subject to a geometric Brownian motion. You go to Geometric Brownian motion and see there that this corresponds to the lognormal value distribution.

 
Colleagues, here's a question I want to raise.
I have long built my trading theories on correlation.
On the correlation dance of euro and pound in relation to each other.
More precisely, I did this by looking at charts of EURUSD and GBPUSD.
Until it suddenly struck me that for n bars of some tf,
EURUSD and GBPUSD charts and, say, EURJPY and GBPJPY have DIFFERENT
correlation coefficients (we are talking about Pearson's linear correlation coefficient).
This, when you think about it, is quite obvious.
But then the question arises in full force - how to calculate something which would describe the correlation of EUR and GBP, and not "eurodollar" and "pounddollar", because the latter obviously does not make any sense.

 
mikfor:
Until it suddenly struck me that, say, for n bars of some tf,
EURUSD and GBPUSD, and, say, EURJPY and GBPJPY have DIFFERENT

correlation coefficients (i.e. Pearson's linear correlation coefficient).

This, if you think about it, is quite obvious.

Quite right, the QCs of {EURUSD; GBPUSD} and {EURJPY; GBPJPY} are different, of course:

This is one of the reasons why the Pearson linear correlation coefficient reading was unflattering.

But then the question arises in full force - how to calculate NOTHING that would describe the correlation of EUR AND GBP, rather than "eurodollar" and "pounddollar", since the latter obviously makes no sense.

There is already an implemented method for not two, but three, four or more financial instruments:

The blue circles show the corresponding linear relationships. The discrepancies of the absolute values are caused by errors in the closing price determination.

Although this is better, it is also bad, because it is not perfect:

Ideally, the sum of the absolute values of the coefficients, rather than the sum of squares, should beequal to one.

If you solve the Recycle method with such an ideal condition, then it will work for two fintechs as well.

 
lea:

hrenfx, have you tried building the scatter plot of those two rows after which you decided to create this thread? ;)

I haven't, but I did for this case of zero correlation:

After reducing MO to zero and variance to one (QC does not change) it looks like this:

Files:
nullcorrelation.rar  7 kb
 
Vinin:

That's pretty clear. I usually use a percentage of the price change. I just wanted to know about the price itself. What is it for?
Exactly, in order to work with the percentage and logarithm it. The price changes exponentially, and the logarithm of the price changes linearly.
 
Mathemat:

I have seen the output of this formula. It relies precisely on a lognormal distribution of the price of the underlying option asset. There, among the underlying assumptions, is the assumption that the price of the underlying is subject to a geometric Brownian motion. You go to Geometric Brownian motion and see there that it corresponds to the lognormal value distribution.

It's simpler than that. Black-Scholes, like so much else in econometrics, is based on the assumption of normality. Everyone admits that this is not quite right, but it is very difficult to make a better approximation to reality. The theory of random walk again rests on the normality of the increments. It was easier that way.

Well, lognormality appears simply because everyone works with the logarithm of the price, i.e. not the price but the percentage of profit - returns. It is impossible to compare two assets with prices of 1 cent and $400 each, but it is possible to compare their logarithms, because they will be separated only by a constant. By removing it we obtain, for example, their historical graph on the same scale.

 
Mathemat:
Logarithms are used to explicitly establish that a quantity with a distribution resembling normal has a lower bound of zero.

1. Exactly, but we know that prices are never below 0.

Mathemat:
In deriving the Black-Scholes formula, it is assumed that the price distribution is lognormal, i.e. it is not the price that is normally distributed, but its logarithm.

2. That said, prices are not distributed lognormally. And what's more, the distribution may be different for different instruments, and still not lognormal.

In both cases we see that the logarithm makes no sense. In the first, it is simply unnecessary. In the second, it's the wrong domain.

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