If we knew exactly how the price was moving... - page 2
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avtomat, it seems to me that stationarity is a logical condition, if it is met, one can talk about a reasonable formulation of the problem. In this case, for example, the ACF depends on the difference in the arguments, which should exactly simplify it a lot.
Of course, it is possible to set a problem without stationarity condition, but is it worth the trouble?
P.S. In my first reply I just pointed out to the author of the thread that knowledge of the pdf alone is not enough, because we describe the process, not the distribution.
Well, in general, it is probably possible to formulate the problem that way, but the stationarity requirement turns out to be an insurmountable barrier. And since it is possible to do without it, I think it is worth the trouble :) Moreover, the process is clearly non-stationary, highly non-stationary - so the stationarity requirement strongly narrows the class of models considered, and, as a consequence, the class of admissible solutions.
Somewhere I saw a book with the theory of optimal stochastic control - there were also diphurs with ACF. I can't find it at the moment, but I've been saving it.
I don't really know what I want either, so I'll try to formulate it without using clever terminology.
Let's say we are trading in a completely abstract market in which quotes are generated by a computer. We know for sure that its price difference distribution will not be bell-shaped with thick tails or even classical Gaussian, but, for example, triangular or saddle-shaped (so we have a "market of crooked mirrors") and we know in advance the distribution formula with all its parameters.
We go to trade at such a market and basically all we want is to make as much money as possible. Such artificial market will open tomorrow and will last for N ticks.
The task is to develop such a trading strategy on the basis of a priori knowledge of price distribution function and available history in order to maximize expected payoff of our deposit after N ticks.
If this distribution has an MO other than zero and greater than the trading costs (spread, commission), then it can be traded. Otherwise, an additional study is needed which will result in a distribution with MO other than zero. Any TS is the reduction of price increment distribution to the distribution of transactions with mO+. The deals are incremental prices in some areas plus money management.
If there is no Mo<>0, but the distribution has some differences from the Gaussian one, for example asymmetry, spikes at some levels etc., then we can build a strategy with positive Mo. I.e. actually convert the initial distribution to a distribution with positive MO.
If mo=0 and the distribution is normal, it does not by itself mean that one cannot construct a profitable strategy (reduce to a distribution with mo+) nor does it mean that one can. In short, it means nothing :)))
If there is no MO<>0, but the distribution has differences from the Gaussian distribution, such as asymmetry, outliers at some levels, etc., then we can construct a strategy with positive MO. I.e. actually convert the initial distribution to a distribution with positive MO.
The words "If no Mo<>0" should be understood as Mo=0 ? If so, it would be interesting to know how "one can construct a strategy with positive MO. That is, actually convert the original distribution to a distribution with positive MO. " ? However, without involving concepts such as "outliers at some levels" etc. That is, relying only on the distribution.
For example, we have an asymmetric distribution with mo=0. If it is asymmetric, then we can find a value of sl and tp (cut off a part of the distribution on the left and on the right) at which the new distribution will be with mo different from zero.
Similarly, for some symmetric but non-Gaussian distributions. Purely by varying sl and tp
Are you talking about the distribution of first price differences or something else ?
Too your claim that sl and tp allow such a dashing distribution seems unfounded. To put it mildly. :-)
I do not see any problem in constructing a profitable strategy if the stationary PRV is exactly known. In principle, you don't need a difur to do this, the problem is solved "graphically", so to speak. Approximately as follows:
1. We select on the PDF plot located on one side of the Y+-spread axis, the area under which is greater than 50%+eps (eps-trading costs + planned winnings) - this area will be equal to the probability of winning P. Accordingly, probability of loss Q= 50%-eps.
2. open a trade on each bar to the side, corresponding to our area PRV
3. The size of the lot to trade is chosen considering that the smaller Pv is, the less capital should be risked. A rather simple calculation leads to the result that in terms of maximum profit increase per N trades (let's take as N the number of trades during which the probability Pv is assumed to be approximately equal - this is not too rigid an assumption) the share of equity exposed must be of the order delta=(P-Q)*{E(|c|)^2/E(c^2)}*100%, where с is a relative price increment per 1 bar, E is an averaging operator.
As can be seen, a necessary condition for successful operation of this system is the presence of the aforementioned segment on the SPW chart, which can in principle be met for a function that is asymmetric relative to the ordinate axis. If this condition is met, the expectation of the system will be strictly greater than zero at any time interval, which means that the trader, confident that he has got the exact WPI for the next bar, can browse through catalogues of tropical islands and choose a banana republic as his property... but that's lyricism.
I see no problem in building a profitable strategy if the stationary PRV is known precisely.
Strange reasoning for a person, who probably heard something about martingales and famous theorem about impossibility to build a profitable system on a martingale.
Alexei, what can you say if the "known stationary PRV" is just ordinary white noise (the integral of it is a Wiener process, a martingale)?