Preliminary Remark about Pareto and Log-normal distributions.
It is custom in the economist to model the financial inequalities with the Pareto distribution (see e.g. https://en.wikipedia.org/wiki/Pareto_distribution ) which is essentially a polynomial function. The exact model of the inequalities is even worse and is closer to the log-normal distribution (see e.g. https://en.wikipedia.org/wiki/Log-normal_distribution ) where the severity of the inequalities is modeled with exponential functions.But here in this book we may keep talking about the Pareto distribution which is celebrated term among the economists
It is well know to economist that the probability distribution of the (capitalization) size of business organizations and domestic economies is the Pareto (or a Power) distribution.(Law 7 of compensation or deeper causalities in the 12 laws of post 9) (see also http://en.wikipedia.org/wiki/Pareto_distribution).
It is custom in the economist to model the financial inequalities with the Pareto distribution (see e.g. https://en.wikipedia.org/wiki/Pareto_distribution ) which is essentially a polynomial function. The exact model of the inequalities is even worse and is closer to the log-normal distribution (see e.g. https://en.wikipedia.org/wiki/Log-normal_distribution ) where the severity of the inequalities is modeled with exponential functions.But here in this book we may keep talking about the Pareto distribution which is celebrated term among the economists
It is well know to economist that the probability distribution of the (capitalization) size of business organizations and domestic economies is the Pareto (or a Power) distribution.(Law 7 of compensation or deeper causalities in the 12 laws of post 9) (see also http://en.wikipedia.org/wiki/Pareto_distribution).
As the organizations inherit their relative size to the relative size of volumes of their packets of transactions , and a continuing trend depends mainly on the population size of orders of the organizations , we deduce that the length and duration of spikes and trends follows also a Pareto (or Power) distribution.The same distribution holds for the volume sizes.
Knowing that, it is not difficult to prove that the optimal trading of Pareto-duration distributed trends is one of pyramiding. Therefore even without the position size variations due to random fluctuation adjustments of the minor component of trading, the optimal major component of trading is not that of a buy-and-hold but requires pyramiding or escalation after the Kelly criterion that defines the position size with the conditional probability of success of the trade . And while for the identification of the trend , the largest time frames give the best trends, for pyramiding the highest resolution time frames give the best choice.
We must not conceive the pyramiding as a greedy tactic, but rather a precocious tactic. Because till the end of pyramiding we never exceed the maximum allowed percentage of exposure of the funds , at the worse case scenario of losing. Instead the pyramiding is a gradual build of the position, where we approach the a maximum allowed percentage of loss of funds in the worse case scenario, gradually as we become more confident that the trend goes on, and while at the same time with a trailing it will close if the trends stops.
We must not conceive the pyramiding as a greedy tactic, but rather a precocious tactic. Because till the end of pyramiding we never exceed the maximum allowed percentage of exposure of the funds , at the worse case scenario of losing. Instead the pyramiding is a gradual build of the position, where we approach the a maximum allowed percentage of loss of funds in the worse case scenario, gradually as we become more confident that the trend goes on, and while at the same time with a trailing it will close if the trends stops.
A better word for pyramiding is escalation. We described above that the escalation (pyramiding) due to the on going continuation of the (constant) trend is the optimal solution. But there is also escalation for different reasons than trend continuation. If for example we are uncertain about a trend, and we utilize say 4 indicators to detect it, and the more of them confirm the trend, then the higher is the reliability of the trend. Depending on the degree of reliability of the trend we might want to escalate (pyramid) the position. We call it trend-reliability escalation or pyramiding. We contrast this to the trend-continuation escalation (pyramiding) that we discussed initially. Furthermore, we may want to classify the trend according it is intensity as Low, Medium, and High. If we escalate the position according to if the trend is Low (smaller position size), Medium (higher position size) and High (highest position size), the we call it trend-intensity escalation (pyramiding). As alternative to the trend-intensity we may utilize acceleration-intensity, therefore making the last the of pyramiding, the acceleration-intensity escalation of the positions size and leverage/exposure. All these three types of pyramiding are different, have different rules and different importance and effect.
The trend-intensity escalation or pyramiding is relevant to the percentage allocation of a portfolio of trading of the 4 patterns as in the post 32. But we want to focus here mainly to the trend-continuation pyramiding.
In the trading practice of Bill Williams as it is described in his books, ("Trading chaos" and "New trading dimensions") he is applying a pyramiding from 1 unit to 5 units as trend-reliability escalation , and when the trend is clear and reliable he goes on pyramiding from 5 units, by increasing by 4 units, then 3 units, then 2 units, as trend-continuation escalation.
The trend-intensity escalation or pyramiding is relevant to the percentage allocation of a portfolio of trading of the 4 patterns as in the post 32. But we want to focus here mainly to the trend-continuation pyramiding.
In the trading practice of Bill Williams as it is described in his books, ("Trading chaos" and "New trading dimensions") he is applying a pyramiding from 1 unit to 5 units as trend-reliability escalation , and when the trend is clear and reliable he goes on pyramiding from 5 units, by increasing by 4 units, then 3 units, then 2 units, as trend-continuation escalation.
Here is an argument on which a proof can be based for the optimality of the escalation or pyramiding, when the trend duration follow a probability density distribution with a tail like the Pareto distribution.
Let us assume that the trend lengths follow a Pareto probability density distribution.
We focus on the conditional probability pc that if the trend has already length at least L0 it will continue to length at least L0+dL . Such a conditional probability is a quotient of the probability density that the trend has length already at least L0 and also has length at least L0+dL, thus p(L=L0+dL) and divided by the probability density of the trend having length at least L0; thus pc=p(L=L0+dL)/p(L=L0)=[from the formula of Pareto density]= ((a*(L0+dL)^a)*a*(L0)^(a+1))/ ((a*(L0)^a)*a*(L0+dL)^(a+1))=[we cancel the a's]=
(((L0+dL)^a)(L0)^(a+1))/ (((L0)^a)(L0+dL)^(a+1))=[we cancel exponents]=L0/(L0+dL) .
Now we can see from the last formula that when the L0 in creases the conditional probability of the trend continuing still further a bit more when it already has length L0, also increases! Notice it is not the absolute probability that increases, as the longer trends have smaller probability, but the conditional probability.
Intuitively this might be so because as the finite duration trend get longer and longer, the conditional probability that it will last one more time step get higher, at least due to the relative size of the next step and the duration of the trend so far.
Intuitively this might be so because as the finite duration trend get longer and longer, the conditional probability that it will last one more time step get higher, at least due to the relative size of the next step and the duration of the trend so far.
This means that if at each time step we have to re-decide our trading, the longer the trend hitherto the more probable it will go on, so therefore the larger the lot size we should choose (as the Lot size depends on the conditional probability of the trend continuing.
This is also one of the proofs based on the Pareto distribution, of the law of statistical momentum conservation of the moves in the markets. This is direct consequence of the stable Pareto distributed, inequality of the size of the business organizations, which is inherited on the volumes of transactions and size of the packets of orders of buying or selling, that are realized in repetitive way in the markets.
And after the Kelly criterion (see post 55 and also https://en.wikipedia.org/wiki/Kelly_criterion ) the position size or exposure depends on the conditional transition probability of success of the trade. But this is exactly the pyramiding or escalation practice. OED.
In particular the above arguments suggests a linear increase of the escalation position size over the trend so far, in other words a quadratic increase of the overall position over the trend length so far.
This is also one of the proofs based on the Pareto distribution, of the law of statistical momentum conservation of the moves in the markets. This is direct consequence of the stable Pareto distributed, inequality of the size of the business organizations, which is inherited on the volumes of transactions and size of the packets of orders of buying or selling, that are realized in repetitive way in the markets.
And after the Kelly criterion (see post 55 and also https://en.wikipedia.org/wiki/Kelly_criterion ) the position size or exposure depends on the conditional transition probability of success of the trade. But this is exactly the pyramiding or escalation practice. OED.
In particular the above arguments suggests a linear increase of the escalation position size over the trend so far, in other words a quadratic increase of the overall position over the trend length so far.
Another way to make the same conclusion (of the optimality of pyramiding) is through the Kelly formula of the optimal account percentage to risk in a stop-loss and thus lot size as described in the posts 3, and 32.
From the post 32, we see that the optimal leverage of the trade and thus its lot size too, is given by the formula
OptimalLeverage=(((((TP/SL)*p-(1-p))/(TP/SL))*Equity)/SL*10)*100
which is derived from the Kelly formula
f=(bp-(1-p))/b (see http://en.wikipedia.org/wiki/Kelly_criterion)
Now if from this formula we start the trade with Lot size L and we expect a StopLoss=SL and a TakeProfit=TP1, and when the market reaches at this price level we realize that now the new anticipated TakeProfit=TP2>TP1, then from this formula is easily deducible that, all other being constant, the leverage of the trade also is to increase; in other words the lot size of the trade. Thus pyramiding or escalation.
A simple way to verify the Pareto or Power distribution is to make the histogram of the
High-Low of bars, e.g. daily bars and/or weekly bars and/or monthly bars etc, or the daily volume size.
For a recent presentation on the role of power distributions in forecasting
see the video
http://www.ted.com/talks/didier_sornette_how_we_can_predict_the_next_financial_crisis.html
A simple way to verify the Pareto or Power distribution is to make the histogram of the
High-Low of bars, e.g. daily bars and/or weekly bars and/or monthly bars etc, or the daily volume size.
For a recent presentation on the role of power distributions in forecasting
see the video
http://www.ted.com/talks/didier_sornette_how_we_can_predict_the_next_financial_crisis.html
We conclude that must not conceive the pyramiding as a greedy tactic, but rather a precocious tactic. Because till the end of pyramiding we never exceed the maximum allowed percentage of exposure of the funds , at the worse case scenario of losing. Instead the pyramiding is a gradual build of the position, where we approach the a maximum allowed percentage of loss of funds in the worse case scenario, gradually as we become more confident that the trend goes on, and while at the same time with a trailing it will close if the trends stops.
No comments:
New comments are not allowed.