Both these strategies give the appearance of profitable trading in the form of graphs constantly going up in the strategy tester. On many segments of the quote history, the tester may show Grail-like results even without any filters. In any field of activity and in any business, there is a danger of more informed people being able to enrich themselves at your expense. Forex trading is no exception.
Forex features plenty of such deceptive practices, and these two strategies are the most illustrative and popular proof of this. The first time you use these strategies, you will see that they work on all currency pairs with incredible profit factors and expected payoffs and can even cope with any spreads, so it seems that this algorithm is beyond the market.
This is because they are based on pure math and no logic. Even after so many years, I would still like to find algorithms allowing me to always get profit regardless of the price direction. Mathematicians are generally interesting people. They are able to prove anything with the right equations regardless of whether this is true in reality.
In general, these two strategies exploit the illusion of breakeven to convince you to use them. Both of them work on any currency pair and any period, or rather create the illusion of working to convince you of their simplicity and efficiency. Delving into these strategies, you will sooner or later realize you do not know anything. However, this stage is necessary as this is the only way to start thinking rationally and understand the true nature of the market and what strategies you really need to use.
The order grid was created with the aim of making profit in any market. It does not matter whether it is falling or growing, if the market features a clearly visible movement, then, according to the idea, the grid opens orders using a clever order opening system so that in total these orders gain enough profit at some point to close them all at once.
Let me show this in the images below:. Here I have displayed two options for rising and falling markets, respectively. According to the grid strategy, we should win regardless of which option we get. Those using the grid always say to use pending orders as they are triggered at the best price. This is true, but I believe that market orders are no worse, if only by the fact that you are able to control spreads and slippages at the time of entry.
Besides, you can slightly postpone the entry. However, limit orders are better in this strategy. We have a starting point, relative to which orders are placed. Above this point, we set buy orders with the step of "s", while below it, we set sell orders. If the price reaches them, they are turned into market ones. The image displays open orders based on the specific pricing situation.
There is no point in displaying limit orders here since they remain at the same levels going up and down indefinitely. Only open real orders are important to us since their profit or loss adds up to the total profit or loss, respectively. To ensure profit, some orders should be greater than others by "K" times, i.
We can also calculate the current profit of a position or orders in case of MetaTrader 4 in the same simple manner. Otherwise, we will get a situation when the price immediately moves in a certain direction and we take the profit after the price moves "n" points up or down. Strictly speaking, this ratio can be calculated but it can be easily selected manually.
The possible calculation looks as follows:. Summing up the loss or profit of all orders, we can see that these sums are arithmetic progressions. There is an equation describing the sum of the arithmetic progression using its first and last terms , which is applied here. Besides, using these equations allows defining the profit factor and expected payoff of trading cycles that are to feature open buy and sell positions.
These equations calculate the profit factor and expected payoff of a certain trading cycle rather than the entire graph. Provided that our graph ended at the end point of the cycle, the profit factor will be positive. A cycle is a separate grid. The grid is built, used to the maximum possible extent, positions are closed and a new grid is built.
This is an infinite process in case of the infinite deposit. This is how it approximately looks on the balance curve:. Here I have provided a simplified representation of a grid robot balance graph when running through history. There are a few cycles that the deposit is able to withstand and the graph goes up. But this inevitably ends with a cycle when the deposit is insufficient and all our visible profit goes to the broker.
In terms of math, this is considered an unfinished cycle. The unfinished cycle is always unprofitable and its loss overlaps all profit earned during the cycles that worked to the end. The cycle may also become unfinished due to insufficient amount of orders necessary to continue the grid. All brokers impose limits on the number of simultaneously open orders in the terminal or on a specific pair.
The grid cannot be constructed indefinitely. Even if we assume that we are able to do that, we still eventually get the above outcome. At the end of the article, I will briefly explain why this happens in terms of math. Like the grid, the idea behind the martingale is to win regardless of the market direction.
It is based on the same illusion of eternal profit. If we open an order and it turns out profitable, we simply trade further. As soon as we get a loss, we increase the lot of the next order "n" times relative to the losing position. If our order is profitable, we simply close it and reset the lot back to the initial value. If the order turns out to be losing again, repeat the previous step while increasing the lot "n" times relative to the sum of lots of all losing positions within the cycle.
Repeat till we have a profitable deal again. The last deal in the cycle is always profitable and its profit always covers the loss of losing deals. This is how the graph starts consisting of cycles. Provided that the graph ended with the last cycle, we get a positive expectation and profit factor. It is of no importance how and where we open these orders.
Preferably, these orders should have fixed profit and loss or simply close on fixed stop levels. Here is how the martingale robot balance graph looks like:. As we can see, it is very similar to the grid balance graph since the martingale works in cycles, just like the grid.
The only difference is that it always opens a single order and waits till it is closed to open the next one. Just like in the case of the grid, sooner or later, the deposit becomes insufficient to finish the cycle, all orders are closed and the deposit is wiped out. To ensure profitable cycles, the profit of the last deal should cover the loss of the previous ones:. Here the profit is calculated in your account currency units rather than in points since the system deals with a lot size.
Lots of a specific order are calculated using recursion:. Spreads, commissions and swaps are not considered here but I don't think this is important. If the need arises, the equations can be easily modified, although I do not see the point in that. The martingale equations are similar to the grid ones. SL and TP are the obtained loss and desired profit of an order. To test the above assumptions, let's write a simple grid EA and a simple martingale in MQL5 language to test them and see the results.
I am going to start with the grid. First, add a couple of convenient classes for working with positions to our template:. These two libraries are always present in MetaTrader 5 by default, so there will be no compilation issues.
The first block implements all the necessary grid parameters, while the second one implements the ability to trade a fixed lot in its simplest form. When launching the EA, we will need the verification and recovery of the grid parameters from the previous session in case the operation was terminated incorrectly. This feature is optional but it is better to implement such things beforehand:.
This code is necessary to implement predefined arrays to complete the initial analysis. We are not going to need these arrays afterwards. We will use them only during the initial calculation. To track the current grid status, we need to additional variables displaying the upper and lower prices during the existence of the grid, as well as the starting grid price and the time it was set.
We also need two boolean variables to track or update grid variables during the price movement, as well as for additional attempts to close the grid if the first attempt failed. The functions for closing positions and clearing the remaining limit orders, as well as the predicate function that detects the condition for closing the grid:. I have commented out the last condition in the predicate function. It closes the grid in case the price moves outside the grid. You can use it at will, it does not change anything else.
Now we only need to write the main trading function:. As you can see, the assumptions about the unprofitable cycle have been confirmed. At first, the grid works pretty well, but then comes a moment when the grid is insufficient leading to a losing cycle devastating all the profit.
Trendy market segments usually demonstrate good results, while losses mostly occur on flat segments. The overall result is always a loss since we still have a spread. Now that we have dealt with the grid, let's move on to the martingale EA. Its code will be much simpler.
To work with positions, we will use the libraries applied in the grid EA. There is no point in displaying the code for the second time. Let's consider the inputs right away:. For more simplicity, I have chosen the system, in which positions are closed strictly by stop loss or take profit. The last variable allows us to avoid constantly loading the entire order history, but rather the necessary window only purely for optimization. I believe, other variables are self-explanatory.
In our case, x is the trade number, y is the balance value at closing the trade. Coefficients of an approximating straight are usually found by least squares method LS method. Suppose we have this straight with known coefficients a and b. Finding the straight by LS method means searching for such a and b that SD is minimal. This straight is also named linear regression LR for the given sequence.
We will call this parameter LR Standard Error. Below are values of this parameter for the first 15 accounts in the Automated Trading Championship However, the degree of approximation of the balance graph to a straight can be measured in both money terms and absolute terms. For this, we can use correlation rate. Correlation rate, r, measures the degree of correlation between two sequences of numbers.
This must be remembered. In our case, we have to compare two sequences of numbers: one sequence from the balance graph and the other sequence representing the appropriate points along the linear regression line. Values of balance and points on linear regression. Let's denote balance values as X and the sequence of points on the straight regression line as Y. To calculate the coefficient of linear correlation between sequences X and Y, it is necessary to find mean values M X and M Y first.
For our example, covariance value is 21 Please note that M X and M Y are equal and have the value of 21 It means that the Balance mean value and the average of the fitting straight are equal. The only thing that remains to be done is calculation of Sx and Sy. Remember how we calculated dispersion and the algorithm of LS method.
As you can see they are all related. The found SSD will be divided by the amount of numbers in the sequence - in our case, 36 from zero to 35 - and extract the square root of the resulting value. So we have obtained the value of Sx. The value of Sy will be calculated in the same way.
This is below one, but far from zero. Thus, we can say that the balance graph correlates with the trend line valued as 0. By comparison to other systems, we will gradually learn how to interpret the values of correlation coefficient. At page "Reports" of the Championship, this parameter is named LR correlation. The only difference made to calculate this parameter within the framework of the Championship is that the sign of LR correlation indicates the trade profitability.
The matter is that we could calculate the coefficient of correlation between the balance graph and any straight. For purposes of the Championship, it was calculated for ascending trend line, hence, if LR correlation is above zero, the trading is profitable. If it is below zero, it is losing. Sometimes an interesting effect occurs where the account shoes profit, but LR correlation is negative. This can mean that trading is losing, anyway. An example of such situation can be seen at Aver's.
There is likely no correlation, in this case. It means we just could not judge about the future of the account. We are often warned: "Cut the losses and let profit grow". Looking at final trade results, we cannot draw any conclusions about whether protective stops Stop Loss are available or whether the profit fixation is effective. We only see the position opening date, the closing date and the final result - a profit or a loss. This is like judging about a person by his or her birth and death dates.
Without knowing about floating profits during every trade's life and about all positions as a total, we cannot judge about the nature of the trading system. How risky is it? How was the profit reached? Was the paper profit lost? Every open position until it is closed continuously experiences profit fluctuations. Every trade reached its maximal profit and its maximal loss during the period between its opening and closing.
MFE shows the maximal price movement in a favorable direction. Respectively, MAE shows the maximal price movement in an adverse direction. It would be logical to measure both indexes in points. However, if different currency pairs were traded,we will have to express it in money terms. Availability of many trades resulted in profits, but having large negative values of MAE per trade, informs us that the system just "sits out" losing positions.
Such trading is fated to failure sooner or later. Similarly, values of MFE can provide some useful information. This may be Trailing Stop that we can move after the price if the latter one moves in a favorable direction. If short profits are systematic, the system can be significantly improved. MFE will tell us about this. For visual analysis to be more convenient, it would be better to use graphical representation of distribution of values of MAE and MFE.
If we impose each trade into a chart, we will see how the result has been obtained. Trades distribution on the plane of MAExReturns. Negative value shows negative slope of the fitting line. If you look through other Participants' accounts, you will see that correlation coefficient is usually positive. In the above example, the descending slope of the line says us that it tends to get more and more drawdowns in order not to allow losing trades.
Correlation Profits, MFE is positive and tends to one 0. This informs us that the strategy tries not to allow long "sittings out" floating profits. It is more likely that the profit is not allowed to grow enough and trades are closed by Take Profit. In development of trading systems, they usually use fixed sizes for positions.
This allows easier optimization of system parameters in order to find those more optimal on certain criteria. However, after the inputs have been found, the logical question occurs: What sizing management system Money Management, MM should be applied. Position sizes may also vary according to the current market phase, to the results of the latest several trades analysis, and so on. So the money-management system applied can essentially change the initial appearance of a trading system.
How can we then estimate the impact of the applied money-management system? Was it useful or did it just worsen the negative sides of our trading approach? How can we compare the trade results on several accounts having the same deposit size at the beginning? A possible solution would be normalization of trade results.
Normalization will be realized as follows: We will divide each trade's result profit or loss by the position volume and then multiply by the minimum allowable position size. For example, order BUY 2. Let's divide the result by 2. Let us do the same with results of all trades and we will then obtain Normalized Profits NP. If the difference between parameters is significant, the applied method has likely changed the initial system essentially.
We can benefit even more from normalized trades in measuring of how the MM method applied influences the strategy. It is obvious that increasing sizes of positions 10 times will cause that the final result will differ from the initial one 10 times. And what if we change the trade sizes not by a given number of times, but depending on the current developments? Results obtained by trust-managing companies are usually compared to a certain model, usually - to a stock index.
Beta Coefficient shows by how many times the account deposit changes as compared to the index. If we take normalized trades as an index, we will be able to know how much more volatile the results became as compared to the initial system 0. Thus, first of all, we calculate covariance - cov Profits, NormalizedProfits. For this, we will calculate the mathematical expectation of normalized trades named M NP. M NP shows the average trade result for normalized trades. The obtained result will be then divided by the amount of trades and name D NP.
This is the dispersion of normalized trades. Let's divide covariance between the system under measuring, Profits, and the ideal index, NormalizedProfits cov Profits, NormalizedProfits , by the index dispersion D NP. The result will be the parameter value that will allow us to estimate by how many times more volatile the capital is than the results of original trades trades in the Championship as compared to normalized trades.
This parameter is named Money Compounding in the "Reports". It shows the trading aggression level to some extent. Now we can revise the way we read the table of Participants of the Automated Trading Championship The LR Standard error in Winners' accounts was not the smallest. At the same time, the balance graphs of the most profitable Expert Advisors were rather smooth since the LR Correlation values are not far from 1. The Sharpe Ratio lied basically within the range of 0. The GHPR per trade is basically located within the range from 1.
And how are things with ldamiani's account? Finally, the last column in the above table, Money Compounding, also has a large range of values from 8 to 50, 50 being the maximal value for this Championship since the maximal allowable trade size made 5. However, curiously enough, this parameter is not the largest at Winners.
The Top Three's values are Did not the Winners fully used the maximal allowable position size? Yes, they did. This is a visible evidence of that money management is very important for a trading system. The 15th place was taken by payday. The EA of this Participant could not open trades with the size of more than 1. What if this error did not occur and position sizes were in creased 5 times, up to 5. Would the Participant then take the second place or would he experience an irrecoverable DrawDown due to increased risks?
Would alexgomel be on the first place? His EA traded with only 1. Or could vgc win, whose Expert Advisor most frequently opened trades of the size of less than 1. All three have a good smooth balance graph. As you can see, the Championship's plot continues whereas it was over!
Opinions differ. This article gives some very general approaches to estimation of trading strategies. One can create many more criteria to estimate trade results. Each characteristic taken separately will not provide a full and objective estimate, but taken together they may help us to avoid lopsided approach in this matter. We can say that we can subject to a "cross-examination" any positive result a profit gained on a sufficient sequence of trades in order to detect negative points in trading.
This means that all these characteristics do not so much characterize the efficiency of the given trading strategy as inform us about weak points in trading we should pay attention at, without being satisfied with just a positive final result - the net profit gained on the account. Well, we cannot create an ideal trading system, every system has its benefits and implications. Estimation test is used in order not to reject a trading approach dogmatically, but to know how to perform further development of trading systems and Expert Advisors.
In this regard, statistical data accumulated during the Automated Trading Championship would be a great support for every trader. It also goes to the bottom of why all the mathematics presented in the article is very suspect. Of course the writer does not deny that the analyses on offer rely on a mathematically false premise all. He in fact admits that straight up - when he acknowledges that the market and the results it throws up do not follow the normal curve.
Professor Mandelbrot - the man who invented fractal geometry shows very clearly in his book "The [Mis] Behaviour of Markets" how hopelessly inadequate normal curve based mathematics is insofar as estimating market behavior goes. Just because the alternative is a lot more difficult and mathematically involved does not justify the false rigor employed in using these meaningless [in terms of real prospect assessments] normal curve based metrics.
But the preceding is not my main point and be that as it may - while it is correct that an equation describing the exact nature of the market is yet to be found and may never be found I think that the basis of that truth should lead us away from the normal curve not to it since no matter how much we massage it the normal curve does not describe the market?
Therefore, anyone not explaining why this is so and then immediately offering the normal curve as a substitute does no trader [serious trader that is] any favors. Of course the reason is pretty simple and indeed has everything to do with the fact that in fact the mathematical nature of the market is exactly known to mathematics via the science of complex dynamical systems which retains chaos theory and fractal geometry within that body of knowledge as its exact explanators of market behavior [and clearly the market is destroyed as the author supposed with the discovery forty two years ago of the mathematical nature of markets which is my point].
What is more, Mandelbrot then shows that given the distributions that describe the market it is false knowledge to bury your head in the sand and claim that the normal curve will suffice. It is absolutely vital to understand the nature of the distributions that describe the market and whether or not we have a prescriptive equation is not the issue - but the realization by traders that these distributions are very different from the normal or Gaussian curve and that therefore - much of the testing measures here are as useless really as throwing up a coin heads or tails in terms of telling the tester anything about the robustness of their strategies, and for the simple reason that the normal curve is a misfit and a huge one at that in describing market fluctuations and therefore outcomes.
Nassim N. Attached files Download ZIP.
This could be calculated manually as n is a smaller value. But the above formulas are useful when n is a larger value. Arithmetic progression is a progression in which every term after the first is obtained by adding a constant value, called the common difference d. For finding the sum of the arithmetic series, S n , we start with the first term and successively add the common difference.
We can also start with the n th term and successively subtract the common difference, so,. Thus the sum of the arithmetic sequence could be found in either of the ways. However, on adding those two equations together, we get. The following table explains the difference between arithmetic and geometric progression :.
Arithmetic progression is a series in which the new term is the difference between two consecutive terms such that they have a constant value. Example 3: Find the sum of the first 5 terms of the arithmetic progression whose first term is 3 and 5 th term is A sequence of numbers that has a common difference between any two consecutive numbers is called an arithmetic progression A. The example of A. The common difference is the difference between each consecutive term in an arithmetic sequence.
Example: 2,4,6,8,……. Arithmetic Progression is any number of sequences within any range that gives a common difference. To find the sum of arithmetic progression , we have to know the first term, the number of terms, and the common difference between consecutive terms.
The number of terms in an arithmetic progression can be simply found by the division of the difference between the last and first terms by the common difference, and then add 1. A real-life application of arithmetic progression is seen when you take a taxi. Once you ride a taxi you will be charged an initial rate and then a per mile or per kilometer charge.
This shows an arithmetic sequence that for every kilometer you will be charged a certain fixed constant rate plus the initial rate. The 'n th ' term in an AP is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next.
To find d in an arithmetic progression, we take the difference between any two consecutive terms of the AP. When the number of terms in an AP goes to infinity, we call it an infinite arithmetic progression. For example, 2, 4, 6, 8, 10, Learn Practice Download.
Arithmetic Progression An arithmetic progression AP is a sequence where the differences between every two consecutive terms are the same. What is Arithmetic Progression? Arithmetic Progression Formula 3. Common Terms Used in Arithmetic Progression 4. General Term of Arithmetic Progression 5. Sum of AP 6. Example 2: Which term of the AP 3, 8, 13, 18, Solution: The given sequence is 3,8,13,18, Breakdown tough concepts through simple visuals.
Math will no longer be a tough subject, especially when you understand the concepts through visualizations. Arithmetic Progression Questions. P: 1, 4, 7, 10, 13,……, Note: We can use another formula to sum up the finite AP. If a number is added or subtracted to each term of a given AP, then the resulting sequence is also an AP. If each term of an AP is multiplied and divided by a certain number except zero , then the resulting sequence is also an AP. If we add or subtract the two different sequences which are in A.
P, then the resultant sequence is also an AP. For example, 2, 4, 6,. And 1, 3, 5,…….. Are two A. The arithmetic mean is defined as the sum of all the observations divided by the number of observations. If AP have odd number of terms, then the middle term is arithmetic mean of the given AP. Then, 7 is the middle number and it is the arithmetic mean. If AP has an even number of terms, then the arithmetic mean is the half of the sum of two middle numbers.
Let us consider an AP:. Select some terms from a sequence which are at regular intervals, then the resultant terms also forms an A. Consider an AP: 1, 3, 5, 7, 9, 11, 13, 15, 17,………. Note: If a condition and sum of terms of the AP is given, then we consider these AP which is given below:. For example: The sum of 4 terms of an AP is If the difference between the first and fourth terms of an AP is Find the AP. Example: Find the sum of squares of the first 5 natural numbers.
Question 1: Find the 56th term of an AP 3, 6, 9,………. Solutions: AP is 10, 20, 30,…………, The sum of first th terms of an AP is. Question 3: If the 3rd term and 7th term of an AP are 17 and 37 respectively. Find the first term of an AP. Solutions: 3 rd term of an AP is Question 1: Find the 65 th term of the progression 3, 8, 13, 18,………… Question 2: Find the sum of the first 50 even numbers.
Question 3: Find the sum of cubes of the first 12 natural numbers. Arithmetic progression is just a sequence having the same common difference between each successive term. The common difference is the value between each number in an arithmetic sequence. The formula to find the common difference of an arithmetic sequence is. Sign Up for Free Already have an account? Sign In. Open in App Create free Account. Search for:. Get Pass Pass.
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The main purpose of this article is to investigate a speculative trading system with a constant magnitude of return rate. We consider speculative operations. Algorithmic trading is a system that utilizes very advanced mathematical models for making transaction decisions in the financial markets. more · Forex Options. An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is.