FR H-Volatility - page 18
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kniff
Ну методы разнятся, вот в наших методах такие понятия как безарбитражность есть.
I don't think the methods are different. I will try to explain with an example.
In order to shoot down the enemy aircraft, you need to know at time t0, where this plane will be in a certain period of time (this time is determined by the distance between the planes and the speed of a projectile, missile....).
By exaggerating and simplifying the task for the Forex market we can put it this way: knowing the coordinates X0,Y0 (exchange rate) you need to determine the coordinates X1,Y1 (rate in the future) with probability 1.
Suppose for a minute that a monk appears and solve this problem easily and beautifully (as it already happened in history (Laplace)). And he will create an algorithm that will predict the exchange rate accurate to a second and pips.
With this example I want to show that in the task of rate prediction the concept of arbitrability and efficiency does not exist.
Having such an algorithm, I can not perform transactions at all, but sell information (the output of this algorithm) or even publish it on any site as a forecast. The concept of arbitrability can only be applied to TS (Trading System), and their TS are a wagon and a small cart. And the concepts and definitions introduced only interfere with the task of predicting.
P.S. To Stratonovich, I still do not understand what you understand by the future. And most importantly, because of that the solutions you get are wrong. I am attaching a file with a good overview of the models. If you don't mind, at least a couple of pages proving this assertion. By simple example (derivative of velocity = acceleration (V(t)/dt=a(t)), derivative of acceleration a(t)/dt= - alfa*a+n(t)) n(t) - BHP, alfa is constant factor characterizing width of spectrum.
P.P.S. Just do not give as a proof the same phrase as in this file after formula (8.8) you all the same mech.matts with average score 5.0.
kniff
Well the methods are different, our methods have such concepts as arbitrability.
I don't think the methods are different. I will try to explain by example.
In order to shoot down the enemy aircraft, you need to know at time t0, where this plane will be at a certain point in time (this time is determined by the distance between the planes and the speed of a projectile, missile).
By exaggerating and simplifying the task for the Forex market we can put it this way: knowing the coordinates X0,Y0 (exchange rate) you need to determine the coordinates X1,Y1 (rate in the future) with probability 1.
Suppose for a minute that a monk appears and solve this problem easily and beautifully (as it already happened in history (Laplace)). And he will create an algorithm that will predict the exchange rate accurate to a second and pips.
With this example I want to show that in the task of rate prediction there is no concept of arbitrage, efficiency.
Having such an algorithm, I can not perform transactions at all, but sell information (the output of this algorithm) or even publish it on any site as a forecast. The concept of arbitrability can only be applied to TS (Trading System), and their TS are a wagon and a small cart. And the concepts and definitions introduced only interfere with the task of predicting.
P.S. To Stratonovich, so far I have not understood, what you understand by future. And the most important, that because of it the received decisions are incorrect. I enclose a file with not bad review of models. If you don't mind, at least a couple of pages proving this assertion. By simple example (derivative of velocity = acceleration (V(t)/dt=a(t)), derivative of acceleration a(t)/dt= - alfa*a+n(t)) n(t) - BHP, alfa is constant factor characterizing spectrum width.
P.P.S. Just do not give as a proof the same phrase as in this file after formula (8.8) you all the same mech.matts with average score 5.0.
The methods are different, which is caused by the difference in the observed phenomenon. But that's a rubbish, but for the point I have the following: the concept of arbitrability certainly exists whether you like it or not - it is defined in any standard textbook on financial mathematics. A completely different question: whether the arbitrage-free condition in real markets is fulfilled: and from a modelling point of view there is every reason to believe that it is (i.e. it is impossible to get a risk-free return). There is no proof of this fact, because arbitrage-free is an assumption of the model, not a result. Models with this assumption underlie the trillion-dollar industry; models without it do not exist. If you want to, build without, it is a matter of the modeler and his sense of reality. If you get something adequate, it will be interesting.
About Laplace I did not understand a little: I never heard that Laplace was a monk (or you had something else in mind?). On the content part, I consider the question of possibility of deterministic prediction as a matter of faith, it is again a supra-model thing.
I cannot say more clearly than I did last time that it is the market, not the strategy, that is arbitrage. Look at the definition: is it about strategy? No, it is about the market, about the process.
About Stratonovich: I don't want to solve Langevin's equation, but I found in your text exactly the place I was talking about: formula 8.4. As we see there the values of functions b and sigma are taken at the left point of the interval [t_k;t_{k+1}], so the obtained process is consistent with the (let's measure relatively) usual filtering process (information at time t). In the case of the Stratonovich integral whose construction is omitted in the text given by you, values of these points will be taken in the middle of the interval [t_k;t_{k+1}] and that will cause that the resulting process - partial sum will "run ahead" (because at time t we do not know the price at time t+dt/2. Of course, in the final formula it is not visible (Brownian motion is continuous, so we look ahead a little bit, no big deal). But firstly, in the case of discontinuous processes the difference will appear in full glory, with unmeasurability of the limit process on available information, and secondly, even in the continuous case the result is different from the Itovian and in practical checking it is much further from reality than the Itovian (it is also clear how to check: you logarithm the increment and check the drift term). It if to tell strictly what is written after 8.8.
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 quite a simple question. Suppose there is a certain SP for which the SP is known. How to calculate the range of SV values in this sample depending on the number N of samples in this sample?
Yura, Sergey, what do you think about it?
Hi Sergei ! Not so long ago you and I complained that there are no experts in mathematical statistics on the forum, no one to listen to professional opinion. And here is luck, not one, but two at once. Let's hear what the experts have to say about the questions we have had at different times.
Dear kamal and kniff, could you please answer a few questions? Your participation in this thread began rather impetuously, but if you didn't come here just to show non-specialists their place, we will 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 nonrandomness? 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 dissertation was an attempt to formulate a possible criterion, but can one say that a process is or is not arbitrable according to 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 FR or SP does not allow to build any TS on its basis.
3. And quite a simple question. Suppose there is a certain SP for which the SP is known. How to calculate a spread of SV values on this sample depending on number N of samples in this sample ?
1. Do you want arbitrability? The thing is that non arbitrage does not exclude a possibility to earn on the average (as in an example with a coin). The criterion for arbitrage-free is (according to the first fndamental theorem in financial mathematics) the existence of a martingale measure, i.e. a physical equivalent distribution measure, that the price process is . This is a lot of highly specialized words, but in a nutshell: a market is arbitrage-free if the probabilities of events in it can be redefined so that the price process becomes a martingale, but the probabilities of events cannot be zeroed out. Example: a coin flip and a game on it, i.e. if a random walk changes to +1 with probability 0.6 and -1 with probability 0.4, the market generated by a coin flip is arbitrage-free because 0.6 and 0.4 can be rewritten to 0.5 and 0.5 and the process becomes a martingale. That's a bit of a stretch, but I suspect you're not interested in arbitrage-free, but in efficiency, which requires the price process to be a martingale without all the transitions to other measures. Finally, I'll stress what you said: this is a theoretical restatement: it's a long way from a practical test of martingale. The problem is that martingale means that a non-trivial forecast is impossible (trivial forecast - price remains the same as it is now), and it is not possible to check martingale - to check the impossibility of such a forecast - in the general case. Shepherd suggests a specific methodology, but it is clearly impossible to check all possible methods. In general it is desirable to consider it as a law of energy generation: it is impossible to prove it, but accepting it, one can get such far-reaching correct effects in modelling, that everybody considers it correct. By the way, this is a true analogy: using modern financial mathematics to build TC is akin to using physics to build a perpetual motion machine - it is possible in principle, but the law of conservation of energy is an axiom. On the other hand I thought and still think that mathematical thinking system allows much better structuring of observed phenomena.
2. No, knowing the distribution of a random series, it is possible to make predictions about the behaviour of some values (future prices) at other prices (current prices). If it is non-trivial, it is possible to make money on it.
3. Range - i.e. distribution (maximum in the sample - minimum in the sample) ?
kamal
Thank you for your reply. As you have seen yourself it is a question of adequacy of models to the process in question. And since the process at time t is not particularly interesting, but the forecast is important, most likely we should take t+dt/2. And model adequacy should be checked in another way, we should investigate the residual value (difference between the forecast and the price). And it would probably be more correct to solve by two ways and by residual value, say, in the first case it obeys the normal law, and in the second case it doesn't. Throw out the bad solution. Regarding the discontinuities, ITO is dying too. So until you're convinced that ITO is better. Stratanovich preserves physics, Ito does not.
For Laplace, he studied at Benedictine monastery school (maybe not a monk, I admit) http://www.math.rsu.ru/mexmat/polesno/laplas.ru.html and at 17 he came to Paris, and started teaching the smart guys to solve integrals, which they, tearing their hair out, proved impossible to solve (no one knew Laplace Transform then :-)). Well, it was like in our case, they called this curve "mandrigal" :-) and gave it arbitrage properties and said that it cannot be solved :-). (just kidding, of course - but what the hell if we can't).
Yurixx
I can't answer the 1st question as you know my point of view on the presence of arbitrage in the price stream.
I disagree with kamal on the second question. (I think he just didn't understand the question, or I did). If I'm wrong, let him correct me.
Yes it is possible to build a TC. One condition is that the SP must change over time. Let me explain by a simple example, let's assume that SP is subject to the normal distribution law (NZR), before news publication it is noise (may be 0), after the news publication it has a signal (may not be 0). Here is a picture.
We set the threshold, in the picture it is set according to the criterion of ideal observer, areas 2 and 4 are equal (these areas in radar are called probability of false alarm Rlt and probability of missing signal Pps), in statistics (errors of the 1st and 2nd kind).
And there are analogues of this trading system (any TS based on channel breakdown), the probability Plt just defines a false breakdown, in the case of a true breakdown it is 3, the probability of correct detection Ppo. (Eh would be so easy in practice).
I don't understand the third question.
Again I apologize for the somewhat unconstructive start to the conversation, for some reason in forum discussions the interlocutor's positions seem more wrong than they actually are. On the list of questions:
1. Do you want arbitrability? The thing is that non arbitrage does not exclude a possibility to earn on the average (as in an example with a coin). The criterion for arbitrage-free is (according to the first fndamental theorem in financial mathematics) the existence of a martingale measure, i.e. a physical equivalent distribution measure, that the price process is . This is a lot of highly specialized words, but in a nutshell: a market is arbitrage-free if the probabilities of events in it can be redefined so that the price process becomes a martingale, but the probabilities of events cannot be zeroed out. Example: a coin flip and a game on it, i.e. if a random walk changes to +1 with probability 0.6 and -1 with probability 0.4, the market generated by a coin flip is arbitrage-free because 0.6 and 0.4 can be rewritten to 0.5 and 0.5 and the process becomes a martingale. That's a bit of a stretch, but I suspect you're not interested in arbitrage-free, but in efficiency, which requires the price process to be a martingale without all the transitions to other measures. Finally, I'll stress what you said: this is a theoretical restatement: it's a long way from a practical test of martingale. The problem is that martingale means that a non-trivial forecast is impossible (trivial forecast - price remains the same as it is now), and it is not possible to check martingale - to check the impossibility of such a forecast - in the general case. Shepherd suggests a specific methodology, but it is clearly impossible to check all possible methods. Generally it is desirable to consider it as a law of energy generation: it is impossible to prove it, but accepting it, one can get such far-reaching correct effects in modelling, that everybody considers it correct. By the way, this is a true analogy: using modern financial mathematics to build TC is akin to using physics to build a perpetual motion machine - it is possible in principle, but the law of conservation of energy is an axiom. On the other hand I thought and still think that mathematical thinking system allows much better structuring of observed phenomena.
2. No, knowing the distribution of a random series, it is possible to make predictions about the behaviour of some values (future prices) at other prices (current prices). If it is non-trivial, it is possible to make money on it.
3. Range - i.e. distribution (maximum in the sample - minimum in the sample) ?
On the contrary, this is a very constructive conversation starter. :-)
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 FR or ACF of a process can we 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 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 thought, 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 that 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.
Yurixx
Yes it is possible to build a TC. One condition is that the SP should change over time. Let me explain using a simple example, suppose the SP is subject to the normal distribution law (NZR), before news publication it is noise (may be 0), after the news publication there is a signal (may not be 0). Here is a picture.
And there are analogs of this trading system (any TS based on the breakdown of the channel), the probability Plt just defines a false breakdown, in the case of a true breakdown it is 3, the probability of correct detection Ppo. (Eh would be so easy in practice)
The change in SP over time is not a problem. It changes all the time. Mostly 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's possible (probably) to view everything that happens as noise, fluctuations, and consider the entire process stationary.
But let's assume everything as you wrote. What should we do with it?
to Yurixx
...
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 after the first flip.
SK. I am well aware that tick volumes on Foreche are too dependent on the data provider and its filters. But you can try, right?