FR H-Volatility - page 2
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If you can, please, explain these concepts in more detail. Unfortunately I don't know the terminology. I would very much like to understand what kind of BP you are analysing?
We are talking about the most common Zig-Zag. We are trying to understand how the average height of the Zig-Zag kinks relates to the formation step. The graph shows all the height variations and their frequency of occurrence for a pitch H=10 points.
But by the way, there is another relationship for the Wiener process which can be used as an arbitrability criterion. Since the Gaussian distribution has an explicit mean and sko, we have sko/mean = root(pi/2). And this is also true for any H partition parameters. It is interesting to check what we actually have, for example, for that distribution in your picture.
For symmetric FRs it is true: sko=SQRT(Sum[(M-x)^2]/[n-1]), mean=Sum[(M-x)]/n), then sko/mean != root(pi/2).
Explain, what do you mean by that?
As far as I understand, in your formulas M is just the average, i.e. the 1st central moment, and n is the number of elements of x. And these are formulas for determining the cumulus and the mean over n elements, i.e. over the sample. And I mean the limit values for the whole normally distributed sequence {x}.
By the way, I was wrong. I wasn't referring to the average, but to the modulus average. So, for the Gaussian FR, which is supposed to describe the distribution of first differences of one-dimensional Brownian motion, with M=0 and sko>0, the integral of |x| (i.e. the modulus mean) is calculated in analytic form and = sko*root(2/pi). Hence we obtain that ratio.
For a sample, of course, differences are possible. But for numbers like 10^6 ticks, this difference should not be significant. Especially if the ends of this interval are not far apart. But this is only if the process is Wienerian and described by a normal distribution.
By the way, I was wrong. I wasn't referring to the average, but to the modulus average. So, for the Gaussian FR, which is supposed to describe the distribution of first differences of one-dimensional Brownian motion, with M=0 and sko>0, the integral of |x| (i.e. the modulus mean) is calculated in analytic form and = sko*root(2/pi). Hence we obtain that ratio.
For a sample, of course, differences are possible. But for numbers like 10^6 ticks, this difference should not be significant. Especially if the ends of this interval are not far apart. But this is only if the process is Wienerian and described by a normal distribution.
Now everything is correct, even for a sample we have: sk*root(2/pi). But the process is far from having a normal distribution:
and it is not Wienerian at all (a sign-variable correlogram different from zero):
Now everything is correct, even for the sample we have: sko*root(2/pi). But the process is far from normal distribution:
and certainly not Wiener's (a sign-variable correlogram different from zero):
Interesting, so for EURJPY ticks the relation |x|=sco*root(2/pi) holds, but the distribution is different from normal ?
And how do you determine whether it is normal or not ? It would be good to see the normal distribution on the FR graph at the same time.
But it is clear what is going on with the familiarity of the carrelogram. If it is plotted for segments of a zigzag (any), then it is absolutely clear that for the neighboring (and all odd shifts) segments the correlation will be negative, but for all even shifts - positive. But if you plot it for the first differences of ticks, then, I suppose, the picture will be different.
How do you determine whether it is normal or not? It would be good to see on a FR chart at the same time a normal distribution.
Please:
Well, it almost does:
As for the familiarity of the carrelogram, everything is clear. If it is drawn for segments of a zigzag (any), it is clear that for the neighboring (and all the odd shifts) segments the correlation will be negative, but for all the even shifts - positive. But if you build it for the first differences of ticks, I suppose the picture will be different.
I don't understand it here, Yura. I plotted the correlogram for the first tick differences (Zig-Zag has nothing to do with it), showing the relation of the "current" tick to each, further and further away. I can show dependence of correlation coefficient between first differences, formed by counts of n ticks in each:
There's something I don't seem to get. On a logarithmic scale, the normal distribution should look like an inverted parabola (i.e. -x^2). In this picture it looks like a linear relationship (i.e. -x) and in the previous post it looks like a hyperbola (i.e. 1/x). If I don't understand something, correct me.
But if I'm right, then this distribution is not normal either.
As for the correlogram, I get it, I made a mistake. Indeed, such a clear sign-variance is surprising. Although a significant negative value for Lag=1 is clear. Even at that discussion we were convinced of an essential market return, especially at ticks level. And, by the way, I obtained very small values of Hvol for ticks, approximately at 1.40-1.50. The last correlogram shows, as I understand it, that market reversion persists at all levels, but tends asymptotically to zero rather quickly. Agreed ?
The difference between 0.89 and 0.80, in my opinion, is not large, but very large. That's over 10%. Think back to the differences we were getting for Hvol from two. They mainly fell in the range 1.95-2.05. A difference of 10% is 1.80 (which was only for ticks) or 2.20 (which was never observed). So, imho, the difference from the normal distribution this ratio shows successfully. The only question is to what extent its difference from 0.80 in one direction or another can be used as a measure of persistence-antipersistence.
PS
Posted and then saw that you changed the picture and it has an inverted parabola. :-))
The last correlogram shows, as I understand it, that market returns persist at all levels, but tends asymptotically to zero rather quickly. Do you agree ?
I agree! I wish we could learn how to use this BP property effectively.
So, imho, the difference from the normal distribution this ratio shows successfully. The only question is to what extent its difference from 0.80 in one direction or another can be used as a measure of persistence-antipersistence.
Why introduce a new measure of consistency-antipersistence when the ACF does it perfectly well. Or is there something you're not telling us?
I agree! I wish we could learn to use this BP property effectively.
Why introduce a new measure of persistence-antipersistence when the ACF is doing a great job. Or is there something you're not telling us?
The use of this case is a question. For all the simplicity of Shepherd's strategy and its seeming obviousness, I think there are pitfalls in it that we have passed over.
I've built a logarithmic distribution for ticks and for several zigzags and got the same results as you did: for ticks you get a curve similar to a hyperbola, for zigzags - straight lines. So, there is no smell of normal distribution here. I wonder why distributions for ticks and zigzags (built on ticks) are principally different? Because a tick is the same zigzag, only with the smallest value of the parameter H=1.
I did not propose to introduce a new measure, I simply stated that this relation can be used as such. In general, in both physics and mathematics, any problem can be solved in several ways. At the same time, there are more ways, not less reasonable, by which the same problem cannot be solved. Just as the solution of a diphu equation is possible in some coordinates and not in others. I have nothing against ACF, it's just that for me this method is not as familiar as others. Besides, in ACF you have to set a fixed Lag, which will be equal to the number of ticks or bars. This is, so to speak, fixing the window on the abscissa axis. But if we are building a zigzag, each sigment can contain absolutely different number of ticks (bars). It is already a fixation of a window along the ordinate axis, the so-called delta modulation. These two methods differ from each other fundamentally.
However, each has its advantages and disadvantages. Among the advantages of ACF, I would mention the possibility of plotting it as a continuous, relatively smooth function. This is not possible with the zigzag method. Maybe it makes sense to use both. Sort of like the additionality principle of quantum mechanics. :-)
Let's do the following. I will calculate (Hvol-2) and ratio (sko/|x|-0. 80) for all H from H=1 (tick zigzag) to H=50 for EURUSD all ticks of 2006 and for model normally distributed series of 2200000 counts, which we then used for comparison. And you do the same for ACF. We will compare the pictures. At worst, we'll see that the variants are equivalent. At best, that they are mutually complementary.
Come on!
What should I build? - A borehole diagram for the Zig-Zag or for the Kagi partitions for H=1...50. That these are not the same thing is evident from the picture. On it the white zig-zag is the extremums proper, and the blue-red broken line is the cagi-partitioning:
It is clear that correlogram for Zig-Zag is useless to build - it will definitely be sign-variable and tend to 1. Kagi constructions can be interesting...
Then I should do the same for a Wiener process with identical volatility, or for a normally distributed model series with the same correlogram as the real one?
Sorry to be a load. I just don't want to do the wrong thing.
What should I plot?
Sergey, look at what I have done and you will understand everything.
Below are plots of the Hvol and sko/|leg| relation to the H zigzag parameter plotted for the EURUSD 2006 ticks. (1969732 ticks) and SV (2200000 ticks). The calculation is performed for the area of values H=1 ... 50. In fact it is a kagi-partitioning. For bars they may not coincide with a zigzag, but for ticks they should. |leg| is an average value of the zigzag segment length.
For convenience, the difference (Hvol - 2) and the difference (sko/|leg| - root(pi/2)) are plotted in red in order to immediately show the difference from the value Hvol=2 that the H-volatility should take for the non arbitrage market and the difference from the value 1.253314 that sko/|leg| should take for the normal distribution.
The following things can be seen from these graphs.
1. The Hvol for the real data and for the model CB both converge to 2, but from different directions. For tick data and small values of H the difference from 2 is significant. And indeed for small intervals the market returns are significant. I think this is why pips strategies would have a good chance if it were not for spread and broker prohibition.
2. the ratio sko/|leg| will differ from the root(pi/2)=1.253314 for almost all values of H of real data and model series. The only exception is H=1 for the model SV. This suggests that the Kagi partition (I think the Renko partition as well) has a different distribution from the normal distribution even if the original series it is based on is normally distributed. And if it is, then all theories and models relying on a normal distribution are deliberately flawed.
3. It turns out that for real data the average value of a zigzag segment is much closer to the value of sko than for the normally distributed series. Since the sko is a measure of volatility, and hence the risk, the riskiness of the game with real data is less than with normally distributed data. Maybe that is why it is still possible to win on Forex?
But that's not all. Following my nerdiness I decided to make sure that the model series is indeed normally distributed. And was unpleasantly surprised. Sergei, here's the FR for the euro and for that model range. No matter how you turn an inverted parabola for the ticks does not work.
But for euros we get exactly the same curves as you do. May be it is because you intentionally tried to reproduce the characteristics of the real series in this model series? In any case, I would like to see how kagi building and their parameters and phd will behave on normal CB. I, for example, find it very strange to see that the distributions for ticks and for zigzags built on these ticks are fundamentally different from each other.