FR H-Volatility - page 4
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If we speak about a closed system in thermodynamic equilibrium, it is obvious that within this system it is impossible to exist channels (quasi-stationary) of energy transfer from one body (group) to another. It is possible to describe the state of such system, in terms of its macroscopic parameters, by means of FR. In this case, it will be equilibrium. The equilibrium PDF may tend to normal distribution (distribution of molecules of ideal gas with respect to velocities), or may tend to another form, e.g. exponential. It depends on the interaction of bodies in the system. But as soon as in the system appears stationary channel of energy loss or inflow (the system is not closed), the FR is visibly distorted. There appears asymmetry, humps on the slopes. This is a signal to action - one can gravitate to the feeder.
Figure shows the "perturbed" - red, and "not perturbed" FR for ticks of EUR/GBP pair normalized by 1 each. The perturbation in this case was considered as the presence of a positive price jump of more than 1 point per tick.
I'm posting pictures, but I'm not satisfied.
First of all, there is not enough data. 175000 samples is still worse than 2 million.
Secondly, generation with such a careful copying of the parameters of the real series still makes its own corrections. Though there is a parabola here, it is so bent that it even cracks. :-)
What I actually wanted to see. My impression is that if we take a normally distributed series of data, which would reproduce only a slope of a real series so that its FR on a chart would look like a classical parabola, then zigzags with H>=5 will have a completely different distribution. And it all has to do with the market arbitrage. In general, I think it is possible to obtain the PDF of the non arbitrage market in the analytical form. I do not know yet how exactly, but the basic idea is there. The main thing - now it is clear that distribution normality has nothing to do with it. So we need to dig somewhere else. And having such a FR we will have something to compare the real data with.
As for these pictures, imho, excluding some minor details, they confirm the conclusions already drawn.
Sergey, how to make a CB generator which will work according to the given distribution ? I would like to try some model distributions. It's still not clear where the hyperbola on the FR plot of real ticks comes from.
Во-первых, мало данных. 175000 отсчетов все-таки хуже, чем 2 млн.
Sergei, how do you make a CB generator that will work according to a given distribution ? I want to try some model distributions. It is still unclear where the hyperbola on the FR plot of real ticks comes from.
There is a gap with statistics.
I'm pasting available EUR/USD ticks, they are more than 2 million, and I'm pasting a number constructed by the AR-model of 4th order.
As for the synthesis of the Random Number Generator with a given FR, I know that this problem is solved for any smooth function that describes distributions. How to do it I do not know. I think, Yura, this problem is up to you. When I needed to approximate the form of FR of real ticks, I used normal distribution of differences raised to a power - it turned out to be close to reality.
I'm waiting for similar results to those presented by you in the previous post.
Sergei, thanks for the new data. Here are the new pictures.
Thick tails, as you can see, are there. But this is just a word, but in general these pictures cause a lot of thoughts. Mostly pessimistic. But first I want to tell you the main thing I've understood these days.
The main condition for arbitrage-free NE series is symmetricity of FR of the first difference series built on this NE. Symmetry ensures that the probabilities of change in the value of NE by the value of H are equal in any direction and at any values. Hence it is clear that the normal distribution of price increments ensures that the market is arbitrage-free, but the reverse is not true. I think that the long life of the efficient market theory is due to the fact that the real FR is not a normal distribution, but has the property of symmetry.
The real FR for ticks on a logarithmic scale is in the form of a hyperbola and for the normal distribution it is an inverted parabola. Why? It is a consequence of the cagi-construction. Indeed, in the cagi-construction a zigzag built on ticks (i.e. with H=1) has the property that any sequence of increments of the same sign is combined into one segment. Then any tick in the opposite direction reverses the direction of the zigzag. This results in a change of direction every tick by +1 or -1, i.e. the number of small ticks does not change. And when directional movements occur, combining co-directional ticks reduces the number of small ticks and increases the number of ticks for larger values of X. It's great that the number of eu and model ticks are equal, thanks to Sergei. As a result it is possible to compare FR values at the same X. The graphs show that for |X|=1 these values are almost equal, but at |X|>1 the differences rapidly increase. For the same reason the FR Z1 of the model series (i.e. the FR of its tick zigzag) loses the form of a parabola and acquires almost flat areas. And I was picking on Sergei about this. :-(
Unlike a tick zigzag, all other zigzags (i.e. with H>1) are constructed in a completely different way. The zigzag segment is obtained when the price changes by the value H. As the result, it joins many (the greater H is, the greater is the number) ticks, half of which are directed upwards, and the other half (with an accuracy to 1) is directed downwards. There can be no joining of neighbouring segments, and hence there is no redistribution of the number of elements in the FR. As a consequence, all FRs for H>1 have the same form both for real and model data.
Another consequence of what has been understood. The ratio sko/|x|, which the Bolsheviks talked so much about, has nothing to do with arbitrability. It depends on the configuration of FR, but can say nothing about its symmetry. As you can see from the top two graphs, the values of this ratio at X=1 (tick zigzag) for real data are closer to the number characteristic of the normal distribution than to the normal distribution itself. Yeah, you probably shouldn't try to characterize a whole curve by a single number. :-)
A couple more words about the pictures. As far as I understand, the data used corresponds to the period from April 2006 till April 2007. If not, Sergey will correct me. It is the period of a practically undecreasing trend. For the euras it was +1558 points, and for the model series it was -1225 points. In general, if you follow Pastukhov, it should at least somehow show up in the values of H-volatility. However, it does not exceed the zero line anywhere (i.e. Hvol-2<0 everywhere), which would be understandable for a return market, but not for a trend market. For real data a large interval of values of the parameter H (from 22 to 41) can be qualified as arbitrage-free. And in general, real data show much more arbitrage-free than normally distributed model data. :-))
Thus, questions arise: is there any use of H-volatility at all ? is it actually capable of identifying the presence of market arbitrage ? After all, as you know, quite a few people made a lot of money during that period. Even those who stupidly sat on a long position. :-) And H-volatility at the same time shows a dip downwards even on the right edge (i.e. on long periods) and thus calls to play against the trend and not on.
Correction, the EUR/USD tick data corresponds to the period April 2006 - August 2007.
By talking about a trending period in this timeframe, you, Jura, are changing the terms "trend" and "trending market". The latter implies a probable continuation of the movement started by the price, regardless of the sign of the direction (deterministic trend). In this case, we can talk about a directional price movement - a trend (stochastic trend), which can reverse at any time. There is no way to detect it statistically and determine when it ends, which means you cannot make money on it. I do not argue that many traders made money on this movement, but I am sure that the same number and in the same volume, these dough lost.
For the speculator the magnitude of price change is of interest - he makes money on it and, in the first approximation, it does not matter over what period of time this price change takes place. One can argue that if there were an ideal TS (in the sense of the maximum possible long-term arbitrage profitability), its logical unit would analyze the amounts of price changes, without reference to time. Kagi strategy satisfies this requirement. However, the analysis of the average statistical yield of the TS based on this strategy for all imaginable TFs and symbols shows that the average long-term yield does not exceed the brokerage companies' commission per transaction. This leads to a sad idea that at the present stage of the Forex market the long term arbitrage profit is fundamentally impossible. I've already commented on this: 'The theory of random flows and FOREX' (post 7).
In general, the AR model is shown here: The theory of random flows and FOREX (post 6). To find autoregressive coefficients of n-th order, we need to solve a system of linear equations (SLE) by Yule-Walker. Yule-Walker system of autocorrelation coefficients r[i] for first differences of initial BP. The general form of the SLU is:
Where, r[0]=1 always. The algorithm for finding autocorrelation coefficients can be found here: 'Random Flow Theory and FOREX' (last post).
P.S. Bastard!!!
Pardon me. It's the third time I've posted this message. The first time my daughter (she's small) crawled up unnoticed and overloaded the MS. Well, well, I think it happens... And lo and behold, as soon as I've prebooted everything, the message is gone again - backed up the forum engine!!! Now I'm sitting here thinking maybe I shouldn't have to do it all a third time. Well, it's not working.
By talking about a trend period in a given timeframe, you, Jura, are changing the terms "trend" and "trending market". The latter implies a probable continuation of the movement started by the price, irrespective of the sign of the direction (deterministic trend). In this case, we can talk about the directional price movement - a trend (stochastic trend), which can turn around at any time. There is no way to detect it statistically and determine the moment when it ends, which means that you cannot make any profit out of it! I do not argue that many traders have made money on this movement, but I am sure that the same number of people, in the same volume, have lost the money.
You are right, of course. But if we do not know if the market is trending or not, and we only take this section, then how can we distinguish it? What are the criteria? Suppose we consider H-volatility as a precursor to such a criterion. Then, judging by its values, the market is non-trendy, and even vice versa, it is more inclined to return. So we're trying to play with the reversion, but it keeps going in one direction for 1 year and 4 months. Nah, in this situation I prefer to rely on real data rather than theoretical conclusions. In this case the criterion of truth - practice - tells me that reliance on H-volatility is fraught with danger.
In general, if you detach yourself from pure statistics and look at Forex from outside, then you can see the following picture. Forex is a system seeking equilibrium. Because the major processes of Forex are dictated by fundamental processes of the global economy, this equilibrium has rather definite limits. The trendiness, i.e. the desire to continue in either direction once it started, could be realized on the purely speculative market, where the psychology of the crowd is not limited by anything. Any trend there could last long enough. But on Forex the economic framework quickly dampens any speculative impulse. At the same time, the economic processes themselves create their own trends, which are reflected in Forex. That's why forex is largely a return market.
The stock market, on the other hand, is many times more suitable as an example of a trend market. Unlike real money, stocks can only be bought and held. Nothing else can be done with the shares I have in my pocket except to hold them. So the price of a stock actually has little to do with economic processes. That is, it is connected only through psychology. If you convince the crowd that stocks are the best investment tool, everyone rushes to buy them. They were convinced that Microsoft doubles the price every year - that's it, everybody will buy it and sit on it until the price doubles. And in the economy, it does not change who has the shares and at what price. This is why the US stock market boom that ended in 2000 was possible. During that time, many stocks increased in price hundreds (!!!) of times. The NASDAQ index rose 5 times and then fell 4 times. And what, the American economy has multiplied during that time? Or maybe it then shrank by half? Absolutely not, just a few percent. That's the effect of psychology!
But never mind. I wrote the last two paragraphs of the previous post just to get down from heaven to earth, closer to H-volatility. I'd be interested to hear from you on the general thoughts of the preceding paragraphs. And besides, I would like to understand: why do we need FR at all? Even if it were known, what could be done with it that would make it possible to build a strategy? Or is it purely theoretical and not applicable in trading practice at all?
That's a great point! I agree with you even in detail.
I even think that the Central Bank is purposely keeping the market "returning" to the trend hysterical crowd - a kind of negative feedback loop acting as a market stabiliser. Of course, the leverage to actuate the stabilization mechanisms is obvious - monetary intervention. But you have to pay for everything! And if we (speculators) build a model similar to the one the Central Bank uses for its interventions...
So all the CB "moves" are inside the spread and we are out of the loop :-(
As for the applicability of FR for the development of TS, my opinion is that it is unsuitable for this, because it is an integral characteristic of the system and, as a consequence, misses a lot of interesting things from an arbitrage point of view.
As for the applicability of FR for the development of TS, my opinion is that it is not suitable for this, because it is an integral characteristic of the system and, as a consequence, it misses a lot of interesting things from the point of view of arbitrage.
To clarify. Are you talking about integral FR or probability density function? In principle they are both quite "integral", but just to be clear ...
What might be more interesting for arbitrage I guess I know your answer - ACF. If I understand you correctly, I agree with that. I remember you once posted an ACF graph built (if I'm not mistaken) on a kagi-partition. From my point of view, it was a ready-made tool for building a strategy. You didn't have to add much to it - you just had to solve a few quite technical issues, and go ahead. I don't know if you did anything in that direction or not.