Rate of price change, how to calculate - page 3
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It is not all that straightforward. The article from the handbook applies only to differentiable processes, while stochastic processes, i.e. those with a random component, do not formally belong to such processes: the limit dS/dt does not exist, hence there is no derivative. As stated above, the price can "wiggle" at any small interval of time, and we cannot get inside this interval for purely technical reasons.
Therefore I think the branch question has a non-trivial sense.
At the end of the bar we have "traveled distance"(tick volume) and "moved" (Close-Open). That is, we can only get average instantaneous speed and average speed. If on a larger scale, the choice is essentially the same. The question arises though, should we continue to calculate the path at the micro level (by ticks) or does it make sense to redefine the price trajectory somehow?
P.S. My point is that technically we can only get this, and the significance of the resulting numbers will actually always be an unsolvable question :).
That's why I added a second post to my first post, which expands the scope of "velocity ".
In other words, if we need some certainty in calculatingthe "rate of pricechange ", we need to understand that this rate, the derivative of a random process, is itself a random process, and that determinism can only come from estimates of moment functions. Therefore, I would reformulate the question from "how to determine the rate of price change" to "how to estimate the first moment of the derivative". And then you can use the whole apparatus of matstatistics.
http://alnam.ru/book_kma.php, chapter 9
Can we be more specific? We have to decide on one implementation after all.
Can we be more specific? We have to make a decision on a single realisation, don't we?
From all the calculations with bounds, etc., it follows quite simple thing: the first moment (expectation, or deterministic component, so to speak) of the derivative is the derivative of the first moment of the initial process. That is, there is already a furnace to dance from. It remains to correctly estimate the first moment, i.e. the mean value of prices. Generally speaking, to do it accurately for the current moment is theoretically very close to obtaining the grail, so I would leave some skepticism about this possibility. But for past moments there is no problem: in the simplest case, we take MA(n) and shift it backwards by n/2+1 periods (average value of group delay), we obtain our estimate, the first difference from it will be the estimate of derivative, i.e. price speed - but! only for past moments. The closer we get to the present moment, the less the influence of the law of large numbers will be, and thus the more we will allow randomness to affect the result.
Once again, the conclusion is that a velocity estimate (even unbiased) can be obtained at any point, but the closer that point is to the present moment, the greater will be the variance of the estimate.
In other words, if we need any certainty in calculating the "velocity of price change", we should understand that this velocity, the derivative of a random process, is a random process itself and that determinism can only be derived from the estimation of moment functions. Therefore, I would reformulate the question from "how to determine the rate of price change" to "how to estimate the first moment of the derivative". And then you can use the whole apparatus of matstatistics.
Of course, a random process.
But just as any process in nature has some inertia, so the process of price movement is inertial, with a noise environment superimposed on it. This slower inertial process can be regarded as the slow component and the noise superimposed on it as the fast component of a single process. But now the provisions of velocity, acceleration, etc. are quite applicable to the slow component. --- although by nature this component has not become deterministic, in the strict sense, but it is no longer random.
The same extraction operation can also be applied to the fast component --- it allows us to go deeper into the process, to see its structure.
Of course, it is a random process.
But just as any process in nature has some inertia, so the process of price movement is inertial, with a noise environment superimposed on it. This slower inertial process can be considered as a slow component and noise superimposed on it as a fast component of the single process. But now the provisions of velocity, acceleration, etc. are quite applicable to the slow component. --- although by nature this component has not become deterministic, in the strict sense, but it is no longer random.
The same extraction operation can also be applied to the fast component --- it allows us to go deeper into the process, to see its structure.
In fact, the same testes, only from the side.
By the way, the way of evaluation can be different, not only what I wrote above. The main thing to keep in mind all the time: if we're estimating an average at some point in time, in order to apply averaging over time, one has to be sure of ergodicity in the given interval, which is not always the case. For example, in such a period, where there is a news release, the condition of ergodicity, most probably, is not fulfilled, and therefore the time averaging is unsuitable.
From all the calculations with bounds, etc., it follows quite simple thing: the first moment (expectation, or deterministic component, so to speak) of the derivative is the derivative of the first moment of the initial process. That is, there is already a furnace to dance from. It remains to correctly estimate the first moment, i.e. the mean value of prices. Generally speaking, to do it accurately for the current moment is theoretically very close to obtaining the grail, so I would leave some skepticism about this possibility. But for past moments there is no problem: in the simplest case, we take MA(n) and shift it backwards by n/2+1 periods (average value of group delay), we obtain our estimate, the first difference from it will be the estimate of derivative, i.e. the price speed - but! only for past moments. The closer we get to the present moment, the less the influence of the law of large numbers will be, and thus the more we will allow randomness to affect the result.
Once again, the conclusion is that a velocity estimate (even unbiased) can be obtained at any point, but the closer that point is to the present moment, the greater the variance of the estimate will be.
Actually the same testes, only from the side.
By the way, the way of estimation can be different, not only what I wrote above. The main thing is to keep an eye on yourself all the time: if we are estimating an average at some point in time, then to apply time averaging to it, we need to be sure of ergodicity in this section, which is not always the case. For example, in such a period, where there is a news release, the condition of ergodicity, most likely, is not fulfilled, and therefore the time averaging is unsuitable.
We cannot have this certainty in principle - already by virtue of the fact that there is only one realisation of the process. So the notion of ergodicity has no practical value here.