I read an authoritative book on trading strategies and wrote my trading robot. To my surprise, the robot does not bring millions, even trading virtually. The reason is obvious: a robot, like a racing car, needs “tuning”, in the selection of parameters adapted to a specific market, a specific period of time.

Since the robot has enough settings, it is time consuming to sort through all their possible combinations in search of the best, too time consuming. At one time, while solving an optimization problem, I did not find an informed choice of the search algorithm for a quasi-optimal vector of parameters of a trading robot. Therefore, I decided to independently compare several algorithms ...

Brief formulation of the optimization problem

We have a trading algorithm. Input data - price history of the hourly interval for 1 year of observations. Output - P - profit or loss, a scalar value.
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The trading algorithm has 4 custom parameters:

1. Mf is the “fast” moving average period,
2. Ms period of the “slow” moving average
3. T - TakeProfit, the target profit level for each individual transaction,
4. S - StopLoss, target loss level for each individual transaction.

We assign a range and a fixed step of change to each of the parameters, a total of 20 values ​​for each of the parameters.

Thus, we can search for the maximum profit (P) for one parameter on one array of input data:

1. varying one parameter, for example, P = f (Ms), producing up to 20 backtests ,
2. varying two parameters, for example, P = f (Ms, T), producing up to 20 * 20 = 400 backtests,
3. varying three parameters, for example P = f (Mf, Ms, T), producing up to 20 * 20 * 20 = 8,000 backtests,
4. varying each of the parameters, P = f (Mf, Ms, T, S) and performing up to 20 ^ 4 = 160,000 backtests.

For most trading algorithms, however, it takes several orders of magnitude more time to conduct a single test. Which leads us to the task of finding a quasi-optimal vector of parameters without having to search through the whole set of possible combinations of them.


In more detail about the trading robot it is possible to read in the following spoiler:


We consider three algorithms for finding the quasi- optimal value of the FC . For each algorithm, we set a limit of 40 tests (out of 400 possible combinations).

Monte Carlo method

or a random selection of M uncorrelated vectors from among the possible number of sets equal to N. The method is probably the simplest possible one. We will use it as a starting point for later comparison with other optimization methods.

Example 1

the graph shows the dependencies of the profit (P) of our trading robot, trading EURUSD , obtained at the annual segment of the history of hourly price measurements, on the value of the parameter - the period of the “slow” moving average (M). All other parameters are fixed and not optimized.



CF (profit) reaches a maximum of 0.27 at the point M = 12. To ensure that we find the maximum value of profit, we need to spend 20 test iterations. The alternative is to conduct a smaller number of tests of a trading robot with a randomly chosen value of the parameter M in the interval [9, 20]. For example, after 5 iterations (20% of the total number of tests, we found a quasi-optimal vector (a vector, obviously, one-dimensional) parameters: M = 18 with a DF value (M) equal to 0.18:



The remaining values ​​on the graph from our optimization algorithm hid the “fog of war”.

Optimization of one of the four parameters of our trading robot, with fixed values ​​of the other parameters, does not allow us to see the whole picture. Perhaps the maximum GF of 0.27 is not the best value of the indicator, if you vary the value of other parameters?



This is how the dependence of the profit on the moving average period changes for different values ​​of the TakeProfit parameter in the interval [0.2 ... 0.8].

Monte Carlo method: optimization of two parameters

The dependence of the profit of a trading robot on two parameters can be graphically depicted as a surface:



The values ​​of the parameters T (TakeProfit) and M (moving average period) are plotted along two axes, the third axis is the value of profit.

For our trading robot, after conducting 400 tests on a data interval of one year (~ 6000 hourly quotes of euro to US dollar), we obtain the surface of the form:



or, on the plane, where the values ​​of TF (profit, P) are represented by color:



Choosing arbitrary points on the plane, in this example, the algorithm did not find the optimal value, but got close enough to it:



How effective is the Monte Carlo method in finding the maximum of the FIT ? After 1,000 iterations of searching for the maximum of the FF on the source data from the example above, I obtained the following statistics:

• the average value of the maximum of the functional phase found during 1,000 iterations of optimization (40 random vectors of parameters [M, T] of 400 possible combinations) was 0.231 or 95.7% of the global maximum of the functional phase (0.279).

Obviously, in comparing the methods of parametric optimization of trading robots, one sample is not an indicator. But for now this evaluation is enough. We proceed to the next method - the method of gradient descent .

Gradient descent method

Formally, as the name implies, the method is used to search for the minimum of the TF.
According to the method, we choose a starting point with coordinates [x0, y0, z0, ...]. Using the example of optimizing one parameter, this can be a randomly selected point:



with coordinates [5] and TF value equal to 148. Next follow three simple steps:

1. check the values ​​of the GF in the vicinity of the current position (149 and 144)
2. move to the point with the smallest ZF value
3. if such is absent, a local extremum is found, the algorithm is completed

To optimize the CF as a function of two parameters, we use the same algorithm. If earlier we calculated CF in two adjacent points $$$[x_ {i-1}], [x_ {i + 1}]$, now we check 4 points:

$$

$$[x_ {i-1}, y_i], [x_ {i + 1}, y_i], [x_i, y_ {i-1}], [x_i, y_ {i + 1}].$$

The method is definitely good when there is only one extremum in the TF of the test space. If there are several extremes, the search will have to be repeated several times to increase the probability of finding a global extremum:



In our example, we are looking for the maximum ZF. In order to remain within the framework of the definition of the algorithm, we can assume that we are looking for a minimum “minus CF”. The same example, the profit of a trading robot as a function of the period of the moving average and the TakeProfit value, is one iteration:



In this case, a local extremum was found, which is far from the global maximum of FC. An example of several iterations of finding the extremum of the CF by the method of gradient descent, the value of the CF is calculated 40 times (40 points out of 400 possible):



Now let's compare the efficiency of searching for the global maximum of the CF (profit) on our source data using Monte Carlo algorithms and gradient descent. In each case, 40 tests are carried out (calculations of ZF). 1,000 optimization iterations were performed by each of the methods:

	Monte Carlo	gradient descent
the average of the obtained quasi-optimal value of the ZF	0.231	0.200
the obtained value of the maximum FC	95.7%	92.1%

As we can see from the table, in this example, the gradient descent method does its job worse — finding the global extremum of the TF — the maximum profit. But not in a hurry to dismiss it.

Parametric stability of the trading algorithm

Finding the coordinates of the global maximum / minimum CF is often not the goal of optimization. Suppose there is a “sharp” peak on the graph - a global maximum, the value of the CF in the vicinity of which is significantly lower than the peak value:



Suppose we chose the settings of the trading robot that match the found maximum of the ZF. If we slightly change the value of at least one of the parameters - the period of the moving average and / or TakeProfit - the profitability of the robot will fall sharply (will become negative). With regard to real trading, one can, at a minimum, expect that the market in which our robot is to trade will differ significantly from the period of history in which we optimized the trading algorithm.

Consequently, when choosing the “optimal” settings of a trading robot, it is worth getting an idea of ​​how sensitive the robot is to changes in settings in the vicinity of the found extremum point of the ZF.

Obviously, the gradient descent method, as a rule, gives us the FF values ​​in the vicinity of the extremum. The Monte Carlo method, rather, beats on the squares.

In multiple instructions for testing automated trading strategies, it is recommended that, after completing the optimization, check the target indicators of the robot in the vicinity of the parameter vector found. But these are additional tests. In addition, what if the profitability of the strategy falls with a slight change in settings? Obviously, you have to repeat the testing process.

An algorithm that, simultaneously with the search for the extremum of the FC, would allow us to assess the stability of the trading strategy to change the settings in a narrow range relative to the peaks found would be useful. For example, do not directly search for the maximum ZF

$$

$$P = f (m_i, t_i),$$

and the weighted average value, taking into account the neighboring values ​​of the objective function, where the weight is inversely proportional to the distance to the neighboring value (to optimize the two parameters x, y and the objective function P):

$$

$$P '(x_i, y_i) = \ frac {w_i \ times P (x_i, y_i) + w_j \ times P (x_j, y_j) + w_k \ times P (x_k, y_k) + ...} {w_i + w_j + w_k + ...}$$

$$

$$w_j = \ sqrt {(x_j-x_i) ^ 2 + (y_j-y_i) ^ 2}$$

$$

$$w_i + w_j + w_k + ... = 1$$

In other words, when choosing a quasi-optimal vector of parameters, the algorithm will evaluate the “smoothed” objective function:

It was



has become

“Sea battle” method

Trying to combine the advantages of both methods (Monte-Carlo and the gradient descent method), I tried an algorithm similar to the “sea battle” algorithm:

• At first I put several “blows” on the whole area.
• then, in places of “hits”, I open massive fire.

In other words, the first N tests are carried out on random uncorrelated vectors of input parameters. Of these, M best results are selected. In the vicinity of these tests (plus - minutes 0..L to each of the coordinates) another K test is being conducted.

For our example (400 points, 40 tests in total) we have:



Again, let's compare the effectiveness of now 3 optimization algorithms:

	Monte Carlo	gradient descent	“Sea battle”
The average value of the found extremum of the TF as a percentage of the global value. 40 tests, 1,000 iterations of optimization	95.7%	92.1%	97.0%

The result is encouraging. Of course, the comparison was made on one specific data sample: one trading algorithm on one time series of the value of the euro against the US dollar. But, before comparing algorithms on more samples of source data, I am going to talk about another, unexpectedly (unjustifiably?) Popular algorithm for optimizing trading strategies - the genetic optimization algorithm (GA). However, the article was too voluminous, and the GA will have to be postponed for the next publication .