FR H-Volatility - page 20
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to Yurixx
Good to see you and the guys too! :о)
Learning to manage time :o).
I didn't have time to go to an island. Alex bought up the entire pacific. According to my intelligence he is now buying up the rest of the moon's surface and challenging a millionaire cat for the right to be the first to go into space.
Yes, I wanted to finish the joke that we are guaranteed to predict that the coin will not remain hanging in the air. But it changes the gist of it.
To Prival
And what is the steady state of the system with a coin?(P.O.: while I was distracted, respected Kamal already asked) And meanwhile I stand my ground - system's aspiration to a steady state gives no advantage for prognostics and I can find a lot of pseudo pseudo arguments. And it doesn't really matter.
PS: I will not disturb scientific research with my "spam", the topic is about something else
You are a mathematician and, moreover, a statistician, I am a physicist. We have different language and different ways of thinking anyway. Therefore, we can only achieve something in a conversation by first reaching an understanding. So thank you for trying to go deeper into the subject and understand each other.
1. If I understood your explanation correctly, the "physical" meaning of arbitrage-free is that one cannot make a prediction that is better than some intrinsic probability of the process. That is, in the case of the coin you cite, it is impossible to predict a +1 with probability 0.7 or -1 with probability 0.5. If this is true, then this understanding of arbitrage-free is certainly broader than what I imagined. However, since in the market losing and winning are initially considered equal probable, it does not change the matter. It turns out that arbitrage-free and inefficient in this situation are effectively equivalent and both are hindered by martingale. Hence, I am actually interested in the criteria of martingality. And I'm interested in it in terms of evaluating the violation of those criteria in a real process.
Checking for martingality by checking all possible techniques is, of course, impossible. So the focus of my question is different. For example, having a FR or ACF of a process, is it possible to determine if the process is a martingale or not ? Or in a narrower sense - some properties of a process function are a necessary and/or sufficient condition. As, for instance, the continuity of a function is a condition that its first derivative can have discontinuities of at most the 1st kind. And another, quantitative, aspect. Is there a quantitative measure of whether a process is martingale ?
The analogy with the law of conservation of energy is quite appropriate. I would even say more: the physical analogy of non-martingale is the claim that any system, given to itself, tends to occupy a position corresponding to the minimum of its potential energy. So the postulate of a no-arbitrage market is well founded. But the market is an open stochastic system with a nonzero relaxation time. I hope you can see what I mean without being strictly ahead of the curve. :-) And that means that by accepting arbitrability in general we cannot assert it in a local sense. Arbitrariness is constantly violated to a greater or lesser extent, depending on the scale of events. And the market is constantly "correcting" this situation, naturally with some lag. This lag is the only opportunity, from my point of view, to make a non-random profit. That is why I want to understand non-randomness and the process of its violation.
The mathematical system of thought, IMHO, allows you to structure any abstract phenomena and objects. When an analogy with reality is found, it extends to observable phenomena. The physical way of thinking allows structuring real phenomena and finding very non-trivial connections in this world. These approaches are hard to do without each other. But together they have provided mankind with all its achievements in the material sphere.
2. Interesting, so I am missing something. Enlighten me, if possible, as to how it can be done in principle.
3. You got it right, only I wasn't referring to the distribution, just the average of the difference between the maximum on the sample and the minimum on the sample.
1. Well, not really, to be honest. The physical meaning of no-arbitrage is roughly the following: you cannot say anything for sure . Of course, you may say something (the price is higher than zero), but you cannot say anything for sure on which you might earn money. You can't say "the coin is sure to fall in the eagle" "the price is sure to exceed today's level tomorrow" etc. The whole power of science in this case is that this (quite a condition) is enough to estimate any derivative from the price process. In our case when trying to earn on Forex the question of non arbitrage is of little interest, what is interesting is the question of efficiency, i.e. the availability of (even risky) opportunities to earn with positive M.O. In the case of coin - the possibility to bet on the more frequently falling out side. Yes, you may be unlucky and the coin will fall on the other side. But you'll win on average. Not exactly, but on average. So speculator is not interested in arbitrage, but interested in efficiency (impossibility to gain profit even with the risk). And the efficiency condition is the marting, in which everything rests.
How to get ahead of marting? Well it's certainly not a spherical horse in a vacuum, you can always tell from a strictly defined process if it's martingale. The distribution function of the process fully specifies the process and yes, you can tell if the process is martingale by it. If the process is a random walk (the sum of independent s.v.), then a necessary and sufficient condition for martingale is that the mean of these quantities is zero. In general (this definition) a process is martingale - if the mathematical expectation of the value one step forward, given all the information up to the current moment is equal to the current value. Not very constructive, I admit. There is no quantitative measure, the statement "process is a martingale" is like saying "temperature is zero" - strictly speaking it is never zero, it is impossible to check this with error meters, but one can try to understand how close the process is to a martingale (there is still a spread, etc.).
Concerning the non-zero relaxation time and others: it seems we are getting closer to the time-honoured fact that on large timeframes the market is very similar to martingale, while on small ones quite different things come into play (requotes, spread, quotes delay, etc.). As they say in the hedge fund industry, "the winner is not the smartest, but the one with the least ping to the exchange". And this is no joke (leading investment banks make special processors to calculate option prices, etc., so time critical).
2. Well I guess I didn't understand that question, because it's quite simple. So there's a coin with heads dropping 6 times out of 10 and tails dropping 4 times out of 10. Bet on heads and on average you'll be in the black :))) A more complex example: if you see that the price increments are anti-correlated, you trade a counter-trend in the appropriate timeframe, and you're in the money. You probably had something more complicated in mind.
3. Are you interested in technique? I mean having a process distribution you can calculate the distribution of the maxima, and once you have calculated the maxima distribution it's easy to calculate the average. Do the same for the minimum, calculate the difference. That is all.
I apologize for quoting in full, but the mutual understanding which was taking shape is fading fast. Therefore I quote in full, so that there was a chance to return to the point of contact without running far and not to lose the constructive.
1. Where did you get the idea that I was thinking something there about accurate predictions ? I did say"better than some intrinsic probability of the process" in my version of the meaning. That's not 0.6, but say 0.7. And this is precisely the supposed statistical advantage, the obtaining of which is the purpose of constructing the TS. I hope you do not believe that I build TS which is guaranteed to win every time? But this is so, just a digression.
I quite agree with what you said about interest (highlighted it in your post). However, there are two almost identical phrases with the opposite meaning. :-) I have to assume that one of these phrases (exactly - where it talks about earning opportunities ) implies inefficiency?
That's why I shifted the emphasis and wrote in my post "Hence, I'm actually interested in the criteria of martingness". To this you answered quite clearly - "the necessary and sufficient condition for martingale is a zero average of these values. "That to me is a comprehensive answer, so thank you very much. Although I must admit that it is somewhat unexpected - too simple a criterion. But it just needs some time to reflect on it, so that the connection to my own ideas becomes apparent.
2. I do not understand here why there has been an abrupt shift to a coin ? The original question was about using FR or SP to build a strategy. Using price correlation or anti-correlation - this option is understandable, I agree. For some reason it seemed more complicated with FR or SP. However, maybe I'm wrong. Maybe if the mo of price increment=0, i.e. a martingale process, then making a profit is impossible regardless of, for example, the shape of the FR or SP curve, or their characteristics ? And if it's mo <> 0, then one should simply put in the appropriate direction without any wisdom and it's allE possibilities relative to FR and SP ?
3 Exactly the technique. And how can one calculate the process distribution of the maximum ?
1.
a) You are confusing "non-arbitrability" with "efficiency" (Amir has already said this).
b) From the essence of the question, I understand you want to derive a method that will answer your question - "is the market arbitrage-free?", "is it efficient". Don't torture yourself with this question - I'll answer it for you myself. The market is ARBITRAL (you can sometimes buy Gazprom shares on the RTS and sell them on the MICEX for a ruble more. With currency as well - sometimes you see one exchange rate in one ECN and another in another). The market is INEFFECTIVE (the proof is the hedge fund industry, which is blossoming and developing).
c) What you say - arbitrage-free and efficient - are some ABSTRACT first and foremost things. From a model, from a checked notebook. The market - real prices - are not an abstract thing about which you can PROVE or SAY something. You can say with some SERIOUS level of certainty, "having observed this data series, you can say with 95% certainty that it has these and those properties". How to check the market for martingale (even with some confidence interval) - I do not know. And there is no point in doing so. It is not marting, it is not marting. There's nothing to check that, either. You can check things like "I have a series: 1 2 4 -2, which is generated by a random variable Xi. With what probability can I say that the expectation of Xi > 0?" You know what I mean? The main point of my reasoning lies in the question you have to understand - VARIABILITY THEORY and MATHEMATICAL STATISTICS are different things. The REAL MARKET is the subject of matstatistics. And THEORETICAL MODELS are theorists. So, martingality is from the theorist, not from the matstat.
2. there are plenty of ideas - but no general approach that will allow you to stamp profitable TS. Don't look for manna from heaven, trading is hard work. For example you can plot distributions of CB populations, you can plot covariance matrices, you can look at persistence / antipersistence of series, you can shove in a neural network, etcetc. There is no general approach. You cannot write a program - use FR or SP as input, and it will give you the output - the code of a ready-made Expert Advisor in MQL4)))
In this case, the idea of discussing specific ideas is constructive, and I would love to. This would be a good place to remember both theorist and matstat, but don't look for IDEAS with the help of the matstat - they are not there. All models of financial markets - in EFFICIENCY and SECURITY.
Here is an example. The example is real, people have made money.
There is the Bleck-Scholes-Merton formula for the fair price of an option. There is the delta-neutral option hedging algorithm. It's all maths, the same math that makes full use of Stochastic integrals and stuff like that. Next, people have an understanding of all this. And, next, people notice that the options market on, let's say, the RTS index is priced way HIGHER than its fair price (well, people calculate volatility - the option price is directly related to price volatility). So what did they do? Sold a bunch of options and hedged.
Here's a typical example - the idea is not derived from formulas, but the maths is used to its full potential.
If you want to discuss a REAL idea, and not invent perpetual motion, you're always welcome)).
3. I don't understand the question.
Thank you, that's very funny. I read it and thought it was written by a zealot for austerity!
My dear, you have not understood a single question ! I didn't ask anything about the market, I didn't ask anything about trading methods. Moreover, I did not set out my ideas or approaches. The three questions I asked are quite limited to the realm of mathematics and pure abstraction. And I was hoping for an answer from someone with a mastery of that mathematics. And indeed, I received such an answer from your senior comrade on the first question. Learn from him while you still have the opportunity.
In general, you should not think for a man his questions. Not to mention the fact that instead of an answer you start giving marks: this is a misunderstanding, this is a wrong approach. If you don't understand something in a question, the best way is to ask again.
And the third point was quite simple - how to calculate? Oh, well...
How can we discuss constructive ideas with you if you do not understand simple, more or less clearly(kamal understood everything) formulated questions?
But thank you for responding.
PS In your post I have marked in red places that I particularly liked.
Okay, everyone has written a lot, I will answer in order.
1. Well, not really, to be honest. The physical meaning of arbitrage-free roughly is as follows: you can't say anything for sure . Of course, you may say something (the price is higher than zero), but you may not say anything for sure on which you might earn money. You can't say "the coin is sure to fall in the eagle" "the price is sure to exceed today's level tomorrow" etc. The whole power of science in this case is that this (quite a condition) is enough to estimate any derivative from the price process. In our case when trying to earn on Forexthe question of non arbitrage is of little interest, what is interesting is the question of efficiency, i.e. the possibility (albeit risky) to earn with positive M.O. In case of coin - the possibility to bet on the more frequently falling out side. Yes, you may be unlucky and the coin may fall on the other side, but the average winnings will be. Not exactly, but on average. So speculator is not interested in arbitrage, he is interested in efficiency (impossibility to gain profit even with the risk). And the efficiency condition is the marting, in which everything rests.
How to get ahead of marting? Well it's certainly not a spherical horse in a vacuum, you can always tell from a strictly defined process if it'smartingale. The distribution function of the process fully specifies the process and yes, you can tell if the process is martingale by it. If the process is a random walk (the sum of independent s.v.), then a necessary and sufficient condition for martingale is that the mean of these quantities is zero. In general (this definition) a process is martingale - if the mathematical expectation of the value one step forward, given all the information up to the current moment is equal to the current value. Not very constructive, I admit. There is no quantitative measure, the statement "process is martingale" is like saying "temperature is zero" - strictly speaking it is never zero, it is impossible to check this with error meters, but one can try to understand how close the process is to martingale (there is still a spread, etc.).
Concerning the non-zero relaxation time and other: we seem to get to that time-worn fact that on large timeframes the market is very similar to martingale, and on small ones quite different things come into play (requotes, spread, delay of quotes, etc.). As they say in the hedge fund industry, "the winner is not the smartest, but the one with the least ping to the exchange". And this is no joke (leading investment banks make special processors to calculate option prices, etc., so time critical).
2. Well I guess I didn't understand that question, because it's quite simple. So there's a coin with heads dropping 6 times out of 10 and tails dropping 4 times out of 10. Bet on heads and on average you'll be in the black :))) A more complicated example: if you see that the price increments are anti-correlated, you trade a counter-trend in the appropriate timeframe, and you're in the money. You probably had something more complicated in mind.
3. Are you interested in technique? I mean having a process distribution you can calculate the distribution of the maxima, and once you have calculated the maxima distribution it's easy to calculate the average. Do the same for the minimum, calculate the difference. That's all.
I apologize for quoting in full, but the mutual understanding which was taking shape is fading fast. Therefore I quote in full, so that there was a chance to return to the point of contact without running far and not to lose the constructive.
1. Where did you get the idea that I was thinking about accurate predictions? In my version of the meaning I said "better than some intrinsic probability of the process". That's not 0.6, but say 0.7. And this is precisely the supposed statistical advantage, the achievement of which is the purpose of building TS. I hope you do not believe that I build TS which is guaranteed to win every time? But this is so, just a digression.
I quite agree with what you said about interest (highlighted it in your post). However, there are two almost identical phrases with the opposite meaning. :-) I have to assume that in one of these phrases (exactly - where it talks about earning opportunities ) inefficiency is implied ?
That's why I shifted the emphasis and wrote in my post "Hence, I'm actually interested in the criteria of martingness". To that you answered quite clearly - "the necessary and sufficient condition of martingness is a zero average of these values." That's a comprehensive answer for me, so thank you very much. Although I must admit that in some way it is unexpected - too simple a criterion. But it just needs some time to reflect on it, so that the connection to my own ideas becomes apparent.
2. I do not understand here why there has been an abrupt shift to a coin ? The original question was about using FR or SP to build a strategy. Using correlation or anti-correlation of price - this option is understandable, I agree. With FR or SP it seemed more complicated to me for some reason. However, maybe I'm wrong. Maybe if the mo of price increment=0, then there is a martingale process and making a profit is impossible regardless of, for example, the shape of the FR or SP curve, or their characteristics. And if it's mo <> 0, then one should not wisely just put in the appropriate direction and that's all the possibilities relative to FR and SP ?
3. Exactly the technique. And how can having a process distribution calculate the maximum distribution ?
1. That's the point: you don't want to make accurate predictions because you obviously understand that it's hardly possible (the hypothesis of no arbitrage in forex is extremely plausible). Therefore you are not interested in (non)arbitrage, you are interested in (non)efficiency ((non)possibility to earn even with risk), i.e. in martingale. Both times in the text you quoted it is written, I think the meaning is absolutely the same both times.
Regarding the simplicity of the answer - unfortunately this answer is true for independent random variables, while price increments in the market can also be dependent.
2. Well the coin is an example. With price FR you seem to have missed the point: although the distribution of price increments can be studied, it alone cannot fully characterize the process. For a non-Gaussian process the correlation function is not enough. The process is characterizedby all finite dimensional distributions, i.e. all constructions of the "joint price distribution at the moments t1, t2, t3 ..." type which is very cumbersome and cannot be studied adequately from the statistical point of view yet. And since the list of behavior included in the FR of the process is very large, we may get extremely complex constructions in different cases, such as "if the process goes up 5 times by 10 pips and then comes back down, a collapse is highly probable".
3. it depends on the kind of process. It would be better to give an example, if you don't mind.
Yes, and somewhere above concerning Stratonovich integral Prival wrote to me that Integral Ito does not generalize to discontinuous processes. This is, to put it mildly, not true. Ito's integral goes on to semimartingales, which includes all Levy processes, e.g. the notoriously discontinuous Poisson's. The argument with the Stratonovich integral has exhausted its relevance - everything I wanted to say can be verified simply by any description of the construction of the Stratonovich integral. To discuss it further seems to be pounding water in a bucket.
Somehow you've got me all wrong, let's sort it out.
1. That's the point: you don't want to make accurate predictions, because you obviously understand that this is highly unlikely (the hypothesis of no arbitrage in Forex is extremely plausible). Therefore you are not interested in (non)arbitrage, you are interested in (non)efficiency ((non)possibility to earn even with risk), i.e. in martingale. Both times in the text you quoted it is written, I think the meaning is absolutely the same both times.
Regarding the simplicity of the answer - unfortunately this answer is true for independent random variables, while price increments in the market can also be dependent.
My understanding is that the consequence of market efficiency is the impossibility of earning an average income, even at risk. Where this efficiency is disturbed, i.e. market inefficiency arises, there is an opportunity to make money at risk. Where am I wrong?
Yes, undoubtedly the price increments in the market cannot be considered independent. But this is a model, a first approximation. Besides, I was not interested in the market in these questions, but in martingale as a mathematical property. I distinguish between theory and practice.
2. Well the coin is an example. With price FR you seem to have missed the point: although the distribution of price increments can be studied, it alone cannot fully characterize the process. For a non-Gaussian process the correlation function is not enough. The process is characterizedby all finite dimensional distributions, i.e. all the constructs of the "joint price distribution at the moments t1, t2, t3 ..." type which is very cumbersome and cannot yet be studied adequately from the statistical point of view. And since the list of behavior included in the FR of the process is very large, we may get extremely complex constructions in different cases, such as "if the process goes up 5 times by 10 pips and then comes back down, a collapse is highly probable".
This construction is indeed very cumbersome. Too much. And I was naturally referring to a simple FR that is a function of one variable. That's why I was interested in your answer: "knowing the distribution of a random series one can make predictions about the behaviour of some values (future prices) at other prices (current prices). "And I don't really need to relate it to the actual market process at all. I wanted to understand how in principle a FR could be used to build a TS. In the abstract. Can you give an example of constructing a prediction in the elementary case of a single variable function. Even a simple indication of the property of FR that allows it would suffice. But again, I don't need recipes, I just want to understand.
3. it depends on the kind of process. Better by example, if you don't mind.
My understanding is that the consequence of market efficiency is the impossibility of earning an average income, even at risk. Where this efficiency is disturbed, i.e. market inefficiency arises, there is an opportunity to make money at risk. Where am I wrong?
Yes, undoubtedly the price increments in the market cannot be considered independent. But this is a model, a first approximation. Besides, I was not interested in the market, but in ynost as a mathematical property. I distinguish between theory and practice.
This construction is really very cumbersome. Too much. And I was naturally referring to a simple FR that is a function of one variable. That's why I was interested in your answer: "knowing the distribution of a random series one can make predictions about the behaviour of some values (future prices) at other prices (current prices). "And I don't really need to relate it to the actual market process at all. I wanted to understand how in principle a FR could be used to build a TS. In the abstract. Can you give an example of constructing a prediction in the elementary case of a single variable function. Even a simple indication of the property of FR that allows it would suffice. But, again, I don't want recipes, I just want to understand.
No problem. There is a gamma distribution with an integer parameter. It integrates in analytic form. Suppose it represents the SP for some series of SV. I have a sample of N1 values of this series, and another sample of N2 values. I want to compare the range of these samples. To do this I need to estimate their maximum values (since the range of SV variation is from 0 to infinity, the sample minimum does not play a role).2. The point is that models with independent increments are extremely simple and no deep answers can be derived from them. That is in a sense it is the first approximation, yes, but theory is in principle capable of the second as well. And practice demands the second :) . Martingale can also have a complex dependent structure, so the power of the concept is much more than just a process with independent equally distributed increments (Levy process).
3. If you assume independence of the increments, it is really the most winning strategy to bet towards the positive mathematical expectation, high science does not diverge from elementary "muscular" feeling here. I.e. the TS will simply be of the "buy and hold" or "sell and hold" type. Again, the case of independent increments is largely trivial. However, it is trivial in terms of profitability; in terms of risk, there are some meaningful observations there as well. In general, mathematics in money management is much more adequate from the viewpoint of the fact that there are correct and clear algorithms for action.
4. You don't have to be a genius to understand that the average spread is greater in the sample with more members :))) However, you probably didn't mean it. In fact, the general algorithm for solution in the case of independent variables is as follows
a) Find the f.r. of each random variable - F(x) (in our case a gamma distribution).
b) Take G(x) = F^n(x)(F to a certain power, en is the sample size)
c) Integrate in a straight line x dG
The value obtained is the mean of the maxima.
1. You are absolutely right, it is.
2. The point is that models with independent increments are extremely simple and no deep answers can be extracted from them. That is in a sense it is the first approximation, yes, but the theory is in principle capable of the second. And practice demands the second :) . Martingale can also have a complex dependent structure, so the power of the concept is much more than just a process with independent equally distributed increments (Levy process).
3. If you assume independence of the increments, it is really the most winning strategy to bet towards the positive mathematical expectation, high science does not diverge from elementary "muscular" feeling here. I.e. the TS will simply be of the "buy and hold" or "sell and hold" type. Again, the case of independent increments is largely trivial. However, it is trivial in terms of profitability; in terms of risk, there are some meaningful observations there as well. In general, mathematics in money management is much more adequate from the viewpoint of having correct and clear algorithms of actions.
4. You don't have to be a genius to understand that the average spread is greater in the sample with more members :))) However, you probably didn't mean it. In fact, the general algorithm for solution in the case of independent variables is as follows
a) Find the f.r. of each random variable - F(x) (in our case a gamma distribution).
b) Take G(x) = F^n(x)(F to a certain power, en is the sample size)
c) Integrate in a straight line x dG
The value obtained is the mean of the maxima.
So there is complete unanimity on the first question. :-)) Great.
2. I understand, in general terms, what you are talking about, but I also understand that this is beyond my mathematical capabilities and maybe even my more specific understanding. :-(
3. Yes, this view of TC is indeed trivial, you don't need to know the FR to do it, just have the mo. I understood it from the beginning. So the question can be formulated in another way: does explicit knowledge of FR give any advantage compared to the elementary case of knowing mo, sko ? Well, and if so, can it be used in some way.
Example. SP has asymmetry (as opposed to Gaussian, which is symmetric), though still mo=0. Can something be extracted from the shape of the curve or is it pointless ?
But this is interesting: "mathematics in money management is much more adequate from the point of view of the fact that there are correct and clear algorithms of action". Can we discuss these algorithms in more detail? That is, what is meant and where can it be found in an accessible form.
4. I am not interested in a qualitative comparison, but in a quantitative one. It is not a logical condition of TC. :-) To be precise, I want to normalize the spread over a sample so that it does not depend on the size of that sample.
I understand the calculation algorithm, but explain pls,
(a) By "each random variable" is meant that each sample of a SV series is a separate variable which has its own distribution ? This assumes that all such variables have the same distribution F(x) ? If not, what does "every random variable" mean ?
b) What is G(x) ? Why do we have to increase F(x) to the power of n and what does this have to do with the sample maximum ? Sorry, as a physicist I need to understand what I am doing.
Gentlemen, explain to me, the military fool. What do you mean by market efficiency. A few pages ago when discussing this concept with Yurixx and Neutron p.12 you seemed to come to the conclusion that the market (the curve that is on the screen does not have this concept) if you claim it does, please give me a formula on how to calculate it. Otherwise it's nothing.
So you don't have to dig around, here's an excerpt
"You have to understand that 'efficiency' is a philosophical concept and can be looked at from different angles. For example you have two shovels in the corner at home. The first is more efficient than the second (that's if you dig), but if you shovel, the second shovel is better (more efficient)."
You can also understand from a bunch of different angles the efficiency of shovel production or sales.
Once again I want to repeat the question, what is the efficiency of the curve (the price you have on your screen).
With this question I want to make it clear to you that it is not there. The effectiveness (plus the arbitrage concept) can only be considered if you have a trading system, so it (TS) may have this concept, it brings you revenue or not, and the curve (market) absolutely does not care how your TS is organized, it may not exist at all.
Introduced concepts only confuse and do not give you a tool to investigate the "behavior" of the curve (to find patterns in it), and leads to thoughts like this quote "And what is the steady state of the system with a coin? (Pause for a moment, dear kamal already asked) And while I stand my ground - the desire of the system to occupy a steady state gives no advantage for prediction and I can find a lot of pseudo-ideal arguments".
Knowing that the system seeks to occupy a steady state and if, as argued here, the market is a martingale, and the necessary and sufficient condition for martingale is zero mean. It is great (just fantastic), it is easy to eat the entire FOREX market, you can trample on its corpse (because the market will die) and wipe your feet with it.
I will explain this statement in pictures, and it doesn't matter whether the stability point is a maximum or a minimum (just rotate the picture 180). The main thing is that it is stable, i.e. it does not change over time.
Now I want to return to this phrase of mine "probability of falling eagle or eagle in the 4th experiment is 0.5, but the probability of falling a row of 4 eagles is not 0.5, if the system tends to its steady state.
With this phrase I wanted to bring you to the idea, that betting on a slope of 2 is more likely than betting on 3 (4 heads in a row), you can also bet on 1 (4 tails), see figure.
You've seen this game strategy 1000 times already, a regular channel strategy (rotate this chart by 90 and imagine how this signal value behaves over time) thresholds are channel lines (you can support and resistance lines).
Yurixx now I see why everyone wants to reduce the non-stationary flow of quotes, to stationary (can=0, variance=const, etc.) If all these characteristics do not change over time (the flow is stationary) here it is a strategy in the figure, undress anyone and Forex including, even bets do not need to be doubled ;-).
I think it is very important to understand what you are analysing and not to confuse flies with cutlets. For market analysis - use the theory of random processes (maybe there is a better one), and for TS analysis - the theory of decision making.
I have already said nice words about Occam's razor, I will put it differently, in Russian, I take a birch stick and ask if the market has efficiency - write the formula, if you can't write it, I'll wave the stick, and so on until you have a formula or you admit that the curve does not have this property.
I have said before that economists come up with definitions without any offense, but financial mathematicians are scarier.
To kniff
All financial market models are in EFFICIENCY and UNARBITRACY.
Teach me a fool, how to calculate the Efficiency and arbitrage-free market, especially when it is declared to have such charming features (see Fig.). I pledge after that to call the steady state point after that as kniff's point and formula (system of equations, integral ....) as great formula ..., unfortunately I do not know the surname but I will surely bring your favorite cognac, that would know.
The inputtask is a stream of quotes on the output (un)efficiency or (un)arbitrage of the market possessing such remarkable properties.
Gentlemen, please explain to me, the military fool. What do you understand by market efficiency. A few pages ago when discussing this concept with Yurixx and Neutron p.12 seemed to come to the conclusion that the market (the curve that is on the screen does not have this concept) if you say that there is pliz, the formula in the studio how to calculate it. Otherwise it's nothing.
Sergei, there is a process that in principle cannot make money in the long term. I'm talking about the Venusian process obtained by integrating a normally distributed SV with zero MO. So, whatever TS you invent, in this case it is doomed to failure. Even theoretically such a TS cannot be created! Let us call such VR EFFECTIVE. As you see, efficiency is a property of this BP, not of a particular TS. I think the analogy made is transparent and intuitively clear?
to Prival
Since this is my assertion, I'll add a little more. My conclusion is based only on common sense, not on the concepts of "martingale" and "efficiency". Moreover - I don't even know what those concepts mean and moreover - I don't want to know. But this ignorance doesn't bother me at all, just a different approach, a different outlook... :о)