Why is the normal distribution not normal? - page 34
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Please! Will someone look at the distribution of the first differences of the currency indices. I assume it is NOT the same as the pairs. But which one ??
It seems that Cauchy should not be involved, but I can't vouch for his alibi... I'm a little crazy :)
Please! Will someone look at the distribution of the first differences of the currency indices. I assume it is NOT the same as the pairs. But which one ??
It seems that Cauchy should not be involved there, but I can't vouch for his alibi... I'm a little crazy :)
Let's look at the behaviour of the currency indices EURx, USDx and the EURUSD currency pair bottom left. The values of the indices and the pair are "pegged" to 1 on the first count for clarity (this does not affect further estimates of the distributions of the increments in the series of first differences). On the top left, the corresponding distributions for the EPR are shown (EURUSD in red). It can be seen that the assumption of Gaussianity of the indices is not experimentally confirmed.
In general, the idea is interesting, but it seems to me that there is an inaccuracy in its basis due to the assumption of "thick-tailedness" of the ROD distribution obtained by the ratio of two CBs distributed in a Gaussian fashion. The fact is that such a construction has little relevance to the real situation if one considers a currency pair as a ratio of indices. Indeed, a currency pair is not a ratio of two CBs with zero MO, but two integrated CBs with MO=0, and this is a big difference. Take a look at the lower right figure. It simulates the behavior of two integrated CB (rnd1, rnd2) with Gaussian distribution in MPR (analog of the indices) and shows BP found as the ratio of these two series (RND2 is an analogue of the price series). The RND distributions of the corresponding series are shown in the top right-hand figure. As would be expected, no thick-tailedness is observed - the distribution is normal in the RNDs and wider than each of them. All distributions are given in logarithmic scale and the normal distribution corresponds to a parabolic curve form (ln(exp[-x^2])=-x^2).
To summarise, the reason lies in the fact that the indices oscillate around a constant value with small amplitude and, as a consequence, the ratio of the indices is not fundamentally different from the indices themselves.
Let us look at the behaviour of the currency indices EURx, USDx and the EURUSD currency pair bottom left. The values of the indices and the pair are "pegged" to 1 on the first count for clarity (this does not affect further estimates of the incremental distributions in the first-difference series). On the top left the corresponding distributions for the RPR are shown (EURUSD in red). It can be seen that (1) the assumption of Gaussianity of the indices is not experimentally confirmed.
Generally the idea is interesting, but it seems to me that (2) is based on inaccuracy associated with the assumption of thick-tailedness of distribution of PDF obtained by the relation of two CB distributed Gaussian. The fact is that such a construction has little relevance to the real situation if one considers a currency pair as a ratio of indices. Indeed, the currency pair, (3) is not a ratio of two CBs with zero MO, but two integrated CBs with MO=0, and this is a big difference. Take a look at the bottom right figure. It simulates the behavior of two integrated CB (rnd1, rnd2) with Gaussian distribution in MPR (analog of indices) and shows BP found as the ratio of these two series (RND2 is an analogue of the price series). The RND distributions of the corresponding series are shown in the top right-hand figure. As would be expected, no thick-tailedness is observed - the distribution is normal in the RNDs and broader than each of them. All distributions are shown in logarithmic scale and the normal distribution corresponds to a parabolic curve (ln(exp[-x^2])=-x^2).
To summarize, the reason is that the indices oscillate around a constant value with a small amplitude and consequently the ratio of the indices is not fundamentally different from the indices themselves.
1) Taki is not confirmed.
2, 3) Taki has such a case. Cried my Nobel Prize... :) ...But the truth is dearer. You are right.
And still there is something in this idea "to generate and divide". Although, as we can see, something is missing. Let's keep thinking.
Thank you very much for the post, Sergei ! And for the work you've done !
Something is clearing up anyway (imha).
1) Taki is not confirmed.
2, 3) Taki has a case. Cried my Nobel Prize... :) ...But the truth comes first. You're right.
And still there is something in this idea "to generate and divide". Although, as we can see, something is missing. Let's keep thinking.
Thanks for the post, Sergei! And for the work done!
Something is clear anyway (imha).
The increment of the pair is equal to: EUR/USD - (EUR+tEUR)/(USD+tUSD), where EUR and USD are the prices of currencies at t, and tEUR and tUSD are increments for the time t
EUR/USD - (EUR+tEUR)/(USD+tUSD)=(EUR*tUSD - tEUR*USD)/(USD*USD + USD*tUSD)
e.g. you can calculate when tEUR/USD parity=1:1
(tUSD-tEUR)/(1+tUSD)
so you could try to generate 2 series, e.g. HP from one subtract the other and divide by itself.
so you could try to generate 2 rows, e.g. HP from one subtract the other and divide by itself.
What for?
Assuming tUSD<<1, we get the first approximation increment of the pair:
EUR/USD - (EUR+tEUR)/(USD+tUSD)=(tUSD-tEUR)/tUSD=1-tEUR/tUSD=1-tEURUSD
For what?
Assuming tUSD<<1, we get the first approximation increment of the pair:
EUR/USD - (EUR+tEUR)/(USD+tUSD)=(tUSD-tEUR)/tUSD=1-tEUR/tUSD=1-tEURUSD...
it seems that it's not tUSD but (1+tUSD) in the denominator and if tUSD<<1, then you just get the difference tUSD-tEUR. I.e. the increment of the currency ratio equals the difference of their increments.
If we generalize under the condition tUSD<USD, it will still result in the difference in increments, but with weights depending on the EURUSD exchange rate in the t-datum.
That's why if we suppose EUR and USD increments to be independent, then EUR/USD increments will be distributed in the same way as EUR and USD increments. Perhaps modeling of dependencies between two random variables will give the needed distribution properties. But it is hardly needed in practice.
Your idea is very good (in the sense of an idea). But somehow I don't understand the implementation... I'm tired of it. I'll reread it tomorrow and try to comment on it.
I reread it, but I still don't get it. Why do you need all this? What do you want to achieve with this generation? There are quite convincing assurances, that the best model of exchange prices today are GARCH-models. Why all the Cauchy, Levy, normal...
P.S. Imho, it's a total waste to estimate what distribution all available row history has. You have to look for local dependencies...
Good question by the way. Maybe create a thread on whether markets are fair/efficient. :)
Hmm. Interesting how you compare price fairness and market efficiency. I didn't even think of such a connection. You're probably right, the closer the price is to a fair price, the more the market picture will resemble an efficient market model. And to put it simply - martingale.
The original message was that time didn't matter at all. Now there is a horizon... But apart from time value of money, there is also such a thing as opportunity cost.
By "freezing" money for an hour instead of the 10 minutes, we lose the possibility to trade several 10-minute deals in other symbols, thus reducing the profitability of the system. That is, time cannot be ignored. It can be analysed in different ways, but it cannot be ignored.
If we knew exactly where and where the movement would take place, there would be no subject for discussion at all. And having an opportunity to trade other trades, we are not insured to "freeze" money in them as well - it is just an opportunity, and its outcome is not known (in this context - by duration). Of course, it is assumed that all instruments are traded by the same TS and therefore it evaluates opportunities on them equally effectively.