FR H-Volatility - page 19
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to Yurixx
The analogy with the law of conservation of energy is quite appropriate. I would even say more: the physical analogy of non-preservability is the statement that any system, given to itself, tends to take a position corresponding to the minimum of its potential energy.
...
Please forgive me for getting involved, and by the same token my incompetence in neither physics nor mathematics. But somehow I am sure that the property of any system to occupy its potential minimum does not affect its predictability. If you take the coin option, for example, then yes, undoubtedly the system will occupy its potential minimum. But it won't help to determine what will happen after the next flip.
Oh, so many years, so many winters! Why don't you show up so often, sir? You must have the meat grinder up and running, so you have no time to come to the forum? Or are you on the islands all the time? :-))
As for the coin, it will have an effect. Only it is necessary to understand what predictability. For example, the desire of a coin makes it possible to predict its imminent return to my palm :-)
to Yurixx
The analogy with the law of conservation of energy is quite appropriate. I would even say more: the physical analogy of no arbitrage is the statement that any system, given to itself, tends to take a position corresponding to the minimum of its potential energy.
...
Please forgive me for getting involved, and by the same token my incompetence in neither physics nor mathematics. But somehow I am sure that the property of any system to occupy its potential minimum does not affect its predictability. If you take the coin option, for example, then yes, undoubtedly the system will occupy its potential minimum. But it doesn't help to determine what will happen next time and what will happen after a certain number of throws.
The probability in this case is not 0.5.
Please forgive me for getting involved, and by the same token my incompetence in neither physics nor mathematics. But I am somehow convinced that the property of any system to occupy its potential minimum does not affect its predictability. If you take the coin option, for example, then yes, undoubtedly the system will occupy its potential minimum. But it doesn't help to determine what will fall next time and what will fall after any number of throws.
I don't mean to be rude... But you can't be.
Yurixx
Changing SP over time is not a problem. It changes all the time. Most of the time people, on the contrary, want to make it unchanging and are looking for stationarity. That's my physical view of the process though, I look at it as local and dynamic. If you take the entire history from the beginning of the market to its end, it is possible (probably) to consider everything that happens as noise, fluctuations, and consider the entire process stationary.But let's say everything is as you wrote. What to do with it?
The point is that the picture is a stationary case that has two states (a signal exists or does not exist); moreover, noise parameters are also stationary - dispersion = const. Stationary process is one when its characteristics don't change with time. It all depends on sampling depth (array to be processed). That's why many people buy that it's easy to plot a channel (or support and resistance lines, the analog of a channel) on history and find the point where the channel breakdown occurs. In my picture it is exceeding the s.c. Threshold. If people start to understand it, they understand that everything depends on the depth of sampling (quality of its plotting) and this is also true for the statistics. Some people stop here and look for some rules of thumb that find them and build TS that begin to bring profit. And some people go further in their research...
The probability in this case is not 0.5.
I'm sorry, I didn't mean that accurately. The probability of getting a nut or an eagle in the 4th trial is 0.5, but the probability of getting four eagles in a row if the system tends to its steady state is not 0.5.
S.K. is this more correct ? or am I being rude and wrong again ?
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 rest on futility. So I am actually interested in the criteria for severity. And I am interested in assessing whether those criteria are violated in the actual process.
Of course it's impossible to check the validity by checking all possible methods. So the focus of my question is different. For example, having FR or ACF of a process, is it possible to determine if it is a process or not? Or in a narrower sense - some properties of a process function are a necessary and/or sufficient condition. As, for example, the continuity of a function is a condition that its first derivative can have discontinuities of no more than the 1st kind. And another, quantitative, aspect. Is there a quantitative measure that the process is an ohm ?
The analogy with the law of conservation of energy is quite appropriate. I would even say more: the physical analogy of non-arbitrage 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 understand 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 thinking, IMHO, allows you to structure any abstract phenomena and objects. When an analogy with reality is found, it is extended 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 in the sample and the minimum in 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 possibility (albeit risky) 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 may fall on the other side, but the average winnings will be. Not exactly, but on average. So for speculator the absence of arbitrage is not interesting, efficiency is interesting (impossibility to earn even with the risk). And the effectiveness condition is the ytnost, in which everything depends.
How can we check the efficiency? Well it's not a spherical horse in a vacuum, you can always tell from a strictly defined process whether it isa martingale or not. The distribution function of the process completely defines this process and yes, one 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 a zero mean of these quantities. 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 kind of 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 anticorrelated - and you trade a counter-trend on the appropriate timeframe, 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.
The probability in this case is not 0.5.
I'm sorry, I didn't mean that accurately. The probability of getting a nut or an eagle in the 4th trial is 0.5, but the probability of getting four eagles in a row if the system tends to its steady state is not 0.5.
S.K. is this more correct ? or am I being rude and wrong again ?
Yura, Sergei, what do you think about this?
Hi Sergey ! We have some thoughts, but let's wait a bit. Not so long ago you and I complained that there were no experts in mathematical statistics on the forum, no one to listen to a professional opinion. And here's luck, not one, but two at once. Let's listen to what experts have to say about the issues that arouse us at different times.
Dear kamal and kniff, could you please answer a few questions? Your participation in this thread started off rather impetuously, but if you didn't come here just to point out non-specialists in their place, we'll be glad to hear your weighty opinion.
The subject of using statistical methods (in our narrow circle) arose a year ago in a parallel forum. At that time Northern Wind also took part in the discussion. Well, a lot of questions were solved, but I personally have some left that I'd like to formulate.
1. What properties of statistical characteristics of NE series (distribution function, probability density function, ACF or others) derive from its non arbitrage? There is a definition of this concept, but it says little in itself. For example, it says nothing about whether a particular process is or is not arbitrage-free. So, there is still a long way to go from this definition to the practical criteria of arbitrability. Pastukhov's thesis was an attempt to formulate one of the possible criteria. But can one say something about the arbitrability of a process by its FR or SP ? I hope I have explained the point clearly.
2. Suppose there is a series of SP and the probability density function for it is known. Are there any ideas or ways to use this function for TC construction? I'm interested in the principle aspect, because I have an opinion that information contained in PDF or SP does not allow to build any TS on its basis.
3. and a very simple question. Suppose there is a certain SP for which the SP is known. How to calculate a spread of SP values in this sample depending on the number N of samples in this sample?
1.
a) You are confusing "arbitrage-free" with "efficient" (Amir has already said that).
b) From the essence of the question, I understand you want to derive a method that will answer the 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 NON-EFFECTIVE (the proof is the hedge-fund industry, which is blossoming and developing).
c) What you say - arbitrage-free and efficient - are some ABSTRACT first of all things. From a model, from a checked notebook. The market - real prices - are not an abstract thing about which you can DEMAND 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 martingale, it is not martingale. 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 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 a theorist, not a matstat.
2. There are plenty of ideas - but no GENERAL APPROACH that will allow you to stamp out 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 concrete ideas, and not invent perpetual motion, you are always welcome)).
3. I don't understand the question.
Sorry, I didn't mean it accurately. The probability of a eagle or eagle falling in the 4th trial is 0.5, but the probability of 4 eagles falling in a row if the system tends to its steady state is not 0.5.
S.K. Is that more accurate? Or am I being rude and wrong again?
What do you mean "if it tends"? In those terms, no one is tending to anything. Simply, the characteristic of this phenomenon is a constant probability of 0.5 for either side of the coin, and an equal probability for any sequence of flips. If the coin was going somewhere, then this property could be detected and exploited. In my mind, there is no such property here (unlike the market, which in my mind has properties on the exploitation of which the TS should be built).