FR H-Volatility - page 3
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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 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.
Hmm... if we follow your logic, then the behavior of Hvol - 2 and difference sko/|leg| - root(pi/2) should be positively correlated. However, in area of small H, where the difference of FR from normal is most noticeable, we observe the smallest values for sko/|leg| - root(pi/2) - as if the distribution tends to Gaussian.
But for the euro the curves turn out to be exactly the same as yours. Perhaps it is due to the fact that you specifically tried to reproduce the characteristics of a real series in this model series? In any case, I would like to see how kagi-builds and their parameters and FR would behave on normal CB. I, for example, find it very strange to see that the distributions for ticks and for zigzags built on those ticks are fundamentally different from each other.
Everything.
Yura, the fact that the model and real series do not have the same Hvol suggests that we (I) are confused about the data files. Let me build a new series (it will take some time to remember) that will have exactly the same correlogram and volatility on ticks as the real BP. I propose to model EUR/JPY ticks as the most promising pair for arbitrage.
Starting to remember. In the N-th order autoregressive model I used for modelling, the FR of the residuals series was very similar to normal if the nature of the FR of the stochastic term (sigma) was Gaussian distributed. To approximate the PDF of the residuals of the model series to the original one, I set a very exotic form of the stochastic term, so there is no Gaussianity.
For the time being I am laying out a series of ticks for EUR/JPY:
Hmmm... if you follow your logic, then the behaviour of Hvol - 2 and the difference sko/|leg| - root(pi/2) should be positively correlated. however, in the region of small H, where the difference of FR from normal is most noticeable, we observe the smallest values for sko/|leg| - root(pi/2) - as if the distribution tends to Gaussian...
I don't know about the positive correlation between Hvol - 2 and sko/|leg| - root(pi/2). From the look of the graphs it seems to me that Hvol and sko/|leg| are quite different characteristics. If we abstract away from the very first point (tick zigzag) of the model series, sko/||leg| behaves very steadily. Probably it can hardly be used in trading, but Hvol seems to be a more valuable characteristic.
As a result of this research I realized that arbitrage-free is not a consequence of normal distribution. More precisely, there are other FRs for which the SV series are arbitrage-free. Hvol is a suitable characteristic for assessing arbitrage-freeness, but sko/|leg| is not. At best it is suitable for estimating the proximity of the FR to a Gaussian, which by itself is of little value.
Of all the many points of the two sko/||leg| plots only one - the first point for the model series - indicates the normality of the distribution. This is exactly the one that directly refers to the series you generated. For me it was quite natural, you specifically generated a normally distributed SV. So it was a surprise for me to see the FR for this series (graph Z1). This once again shows that sko/|leg| may be a good characteristic to assess the normality of FR, but clearly not exhaustive. :-)
The posted ticks are real data or modelled ?
PS
By the way, I think it is not necessary that correlogram and volatility of model series coincide with real data. Our task, after all, does not yet go beyond a fundamental test of the behaviour of these characteristics. On the contrary, if it is the most primitive normally distributed series, even if they are out of touch with reality, it is even better. But if it becomes clear on such a series that, yes, these characteristics work, then we may ask a second question: may these characteristics distinguish between real and model data (fake :-), may they be a filter of arbitrage possibilities?
Real! Real.
And here come the model ticks!
When generating them, the main condition was the coincidence of correlograms and volatility on different samples:
For this purpose an autoregressive model of the 5th order was used. Here is how BPs themselves and their FRs behave:
As a result of this study, I realized that arbitrariness is not a consequence of the normal distribution. More precisely, there are other PDFs for which the SV series are arbitrage-free. Hvol is an appropriate characteristic for assessing arbitrage-freeness, but sko/|leg| is not. At best it is suitable for assessing the proximity of the FR to a Gaussian, which by itself is of little value.
Seems to me you've highlighted a very important point: Arbitrariness is not a consequence of the normal distribution. I should add that arbitrage may be a consequence of a non-equilibrium in Forex (we are not talking about its type yet).
That's how the autoregressive coefficient values of the model and source coefficients coincide:
P.S. Yura, explain me, how can it be that such important process characteristics as volatility, correlogram!!! coincide, values of autoregressive coefficients and FR differ fundamentally!? Mathemat suggested that it is from non-stationarity in the strict sense in a number of first residuals... but somehow it's not convincing. Shit!
Yes! All the data is for ticks for July this year, that's what was modelled.
The correlogram algorithm itself already implicitly assumes that the process is considered stationary. How do you know that,Neutron?
By the way, the ticks by amplitude are very similar to a stationary process (it's almost always +-1 if it's a Euro). By lag (time between ticks) - not at all.
P.S. Here would be nice to build bars with equal number of ticks, not with equal astronomical time in them...
P.P.S. Here they are, the roots of the probable non-stationarity of bars. We are digging in amplitude, but we should be digging in time... Maybe on such a representation of the process Prival' s ideas will work. What do you think, Neutron?
A number of first tick differences have expectation strictly at zero, the standard deviation varies from session to session, but as you correctly noted - weakly... I think the problem is in the inadequacy of the model used. Indeed, it does not take into account news disturbances that cause growth of "fat tails". If we introduce a term that seldom but aptly "spreads" ticks the picture will become more realistic. But, how much do we need it? Yura has something to say about that...
Erotic somehow :-))
It seems to me that you have stressed a very important point: Arbitrariness is not a consequence of normal distribution. I would add that arbitrage may be a consequence of an imbalance in the FR (we are not talking about its type yet).
P.S. Yura, explain me, how can it be that such important process characteristics as volatility, correlogram!!!, values of autoregressive coefficients coincide and FR differ fundamentally! Mathemat has suggested that this is from a lack of stationarity in the strict sense in a number of first residuals... But, somehow it's not convincing.
What is non-equilibrium FR ? And what is stationarity in the strict sense ? Well don't forget that I'm not a mathematician. :-) By the way, I picked up Landau-Lifshitz volume "Statistical Physics" yesterday, found so many interesting things there ! That's when I bitterly regretted that I had studied fields and not statistics. :-))
Honestly, I can't answer the question. I myself am still perplexed by everything I have seen in the last couple of days. I have downloaded the data, but have not counted it yet, give me time.
Sergey, I think you were absolutely right when talking about generalized exponential distribution. From the looks of it, it really is something like that. And one more thing I want to agree with you completely. This one:
I think the problem is the inadequacy of the model used. Indeed, we don't account for news disturbances in it, which is where the "fat tails" proliferate. If we introduce a member who will seldom but aptly "spread" ticks, the picture will become more realistic.
Apart from that, there is another working idea. I would like very much to see the FR and all characteristics of a real series during a pronounced, stable trend. One problem - trends don't happen for such a long time that the amount of data is sufficiently representative. Or maybe I don't understand something? May be it's possible to cut pieces and combine them into one series? In general I don't know how to do it, but I really want to look at FR in different states of the market. After all, what we are actually looking at is the average hospital temperature.
P.S. It would be nice to build bars with an equal number of ticks rather than equal astronomical time in them...
It's generally not difficult at all. I can do that too and post the relevant statistics, just tell me which one. Also, are you only interested in returns or ONLC ? I think Northwind did something like that.
But I can't agree that this is the root of non-stationarity. I do not know what is stationary yet (but I hope you will write it), but I suppose forex cannot be stationary in any case. But quasi-stationary may be. Any way you look at it, forex is a stable, stable system. It absorbs and dissipates any external disturbances. That is Forex as a system sits in a deep well (potential well, of course, excuse the pun :-) And if one throws a brick into this well that waves on the water will be guaranteed, but the equilibrium will be restored. So all models based on stationarity are quite entitled to life. With one significant "but":
If it can be shown that all phenomena that disturb the stationarity of forex have little effect on the statistical parameters of the process. And if the opposite can be shown, then in the process it will probably be possible to determine where and how non-stationarity occurs. And then the question will be solved what traders earn: stationarity or lack of it.
By the way Mathemat, you once wrote about risks and the influence of the fact that the cramp is larger than the average. Maybe you can comment on the result: for real price data the difference between sko and average is much smaller than for normally distributed SV.
Yurixx, stationarity comes in two senses - broad and narrow.
The broad sense is when the r.O. of the process is constant and the ACF depends only on the difference of the arguments and not on each of them separately. Probably, by "constancy" of the r.O. you mean stationarity again :) That's a strange definition to get...
In the narrow sense it is when... Forget about it in the narrow sense. This stationarity is practically unverifiable.
"Prices are an unsteady response to an unobservable stationary sequence", (c) unknown. This point of view has been very close to me lately: there is a God who observes the original "good" sequence, but for mere mortals he passes it through some non-linear filter to make it non-stationary.
Honestly, I don't recall. I remember writing that risks are affected by non Gaussian distribution (thick tails).