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### Stock Market Math: Essential Concepts for Algorithmic Trading,Who is a Quant or Quantitative Analyst?

All functions here are intended either for calculating the values of arrays, or they implement some auxiliary mathematical functions, except for the first two.

They are called during initialization along with the calculation of the neutral distribution, and used to set the size of the arrays. Next, create the code block for calculating the real distribution and its main parameters in the same way. Here all is simple but there are much more arrays since the graph is not always mirrored relative to the vertical axis.

To achieve this, we need additional arrays and variables, but the general logic is simple: calculate the number of specific case outcomes and divide it by the total number of all outcomes.

This is how we get all probabilities ordinates and the corresponding abscissas. I am not going to delve into each loop and variable.

All these complexities are needed to avoid issues with moving values to the buffers. Here everything is almost the same: define the size of arrays and count them. Next, calculate the alpha and beta trend percentages and display them in the upper left corner of the screen.

CurrentBuffer and NeutralBuffer are used here as buffers. For more clarity, I have introduced the display on the nearest candles to the market. Each probability is on a separate bar. This allowed us to get rid of unnecessary complications. Simply zoom the chart in and out to see everything. The CleanAll and RedrawAll functions are not shown here. They can be commented out, and everything will work fine without rendering. Also, I have not included the drawing block here.

You can find it in the attachment. There is nothing notable there. The indicator is also attached below in two versions — for MetaTrader 4 and MetaTrader 5. I have developed and seen plenty of strategies.

In my humble experience, the most notable things happen when using a grid or martingale or both. Strictly speaking, the expected payoff of both martingale and grid is 0.

Do not be fooled by upward-going charts since one day you will get a huge loss. There are working grids and they can be found in the market. They work fairly well and even show the profit factor of This is quite a high value. Moreover, they remain stable on any currency pair. But it is not easy to come up with filters that will allow you to win.

The method described above allows you to sort these signals out. The grid requires a trend, while the direction is not important. Martingale and grid are the examples of the most simple and popular strategies. However, not everyone is able to apply them in the proper way. Self-adapting Expert Advisors are a bit more complex.

They are able to adapt to anything be it flat, trend or any other patterns. They usually involve taking a certain piece of the market to look for patterns and trade a short period of time in the hope that the pattern will remain for some time. A separate group is formed by exotic systems with mysterious, unconventional algorithms attempting to profit on the chaotic nature of the market.

Such systems are based on pure math and able to make a profit on any instrument and time period. The profit is not big but stable. I have been dealing with such systems lately. This group also involves brute force-based robots. The brute force can be performed using additional software. In the next article, I will show my version of such a program.

The top niche is occupied by robots based on neural networks and similar software. These robots show very different results and feature the highest level of sophistication since the neural network is a prototype of AI. If a neural network has been properly developed and trained, it is able to show the highest efficiency unmatched by any other strategy.

As for arbitration, in my opinion, its possibilities are now almost equal to zero. I have the appropriate EAs yielding no results. Someone trades on markets out of excitement, someone looks for easy and quick money, while someone wants to study market processes via equations and theories.

Besides, there are traders simply having no other choice since there is no way back for them. I mostly belong to the latter category.

With all my knowledge and experience, I currently don't have a profitable stable account. I have EAs showing good test runs but everything is not as easy as it seems. Those striving to get rich quickly will most probably face the opposite result. After all, the market is not created for a common trader to win.

It has quite the opposite objective. However, if you are brave enough to venture into the topic, then make sure you have plenty of time and patience. The result will not be quick. If you have no programming skills, then you have practically no chance at all.

I've seen a lot of pseudo traders bragging about some results after having traded deals. In my case, after I develop a decent EA, it may work one or two years but then it inevitably fails In many cases, it does not work from the start. Of course, there is such thing as manual trading, but I believe it is more akin to art. All in all, it is possible to make money on the market, but you will spend a lot of time. Personally, I don't think it is worth it.

From the mathematical perspective, the market is just a boring two-dimensional curve. I certainly do not want to look at candles my entire life. I believe that the Grail is more than possible. I have relatively simple EAs proving it. Unfortunately, their expected payoff barely covers the spread. I think almost every developer has strategies confirming this. The Market has plenty of robots that can be called Grails in all respects. But making money with such systems is extremely difficult as you need to fight for each pip, as well as enable spread return and partnership programs.

Grails featuring considerable profits and low deposit loads are rare. If you want to develop a Grail on your own, then it is better to look towards neural networks. They have much potential in terms of profit.

Of course, you can try to combine various exotic approaches and brute force, bit I recommend delving into neural networks right away. Oddly enough, the answer to the questions of whether a Grail exists and where to look for one is quite simple and obvious to me after tons of EAs I have developed.

The first point is the most important here. If you have a profitable strategy regardless of whether it is manual or algorithmic , you will always want to intervene.

This should not be allowed. Situations, in which profitable deals are less numerous than losing ones, exert a considerable psychological impact ruining a trading system. Most importantly, do not rush to win back your losses when you are in the red.

Otherwise, you may find yourself with even more losses. Remember about an expected payoff. It does not matter what the current position's equity loss is. The next important thing is a lot size you apply in your trading. If you are currently in profit, make sure to gradually reduce the lot.

Otherwise, increase it. However, it should be increased only up to a certain threshold value. This is a forward and reverse martingale.

If you think carefully, you can develop your own EA based purely on lot variations. This will no longer be a grid or martingale, but something more complex and safe. Besides, such an EA may work on all currency pairs throughout the history of quotes. This principle works even in a chaotic market, and it does not matter where and how you enter. With proper use, you will compensate for all spreads and commissions, and with masterful use, you will come out with a profit even if you enter the market at a random point and in a random direction.

To reduce losses and increase profits, try to buy on a negative half-wave and sell on a positive half-wave. A half-way usually indicates the previous activity of buyers or sellers in the current market area, which in turn means that some of them have been market ones, while open positions will close sooner or later pushing the price in the opposite direction.

That is why the market has a wave structure. We can see these waves everywhere. A purchase is followed by a selling and vice versa. Also close your positions using the same criterion. Everyone's perspective is subjective. In the end, it all depends on you, one way or another. Despite all the disadvantages and wasted time, everyone wants to create their own super system and reap the fruits of their determination.

Otherwise, I do not see the point of delving into Forex trading at all. This activity somehow remains attractive to many traders including myself. Everyone knows how this feeling is called, but it will sound childish. Therefore, I will not name it to avoid trolling. Translated from Russian by MetaQuotes Ltd. You agree to website policy and terms of use.

Do you like the article? Share it with others — post a link to it! Use new possibilities of MetaTrader 5. Similar articles Data Science and Machine Learning Part 09 : The K-Nearest Neighbors Algorithm KNN Developing a trading Expert Advisor from scratch Part 29 : The talking platform How to deal with lines using MQL5 Neural networks made easy Part 26 : Reinforcement Learning Developing a trading Expert Advisor from scratch Part 28 : Towards the future III.

Introduction I am a developer of automatic strategies and software with over 5 years of experience. Why is it so challenging to find entry and exit points? Market mechanisms and levels Let me tell you a little about pricing and powers that make the market price move. Mathematical description of the market What we see in the MetaTrader window is a discrete function of the t argument, where t is time.

To introduce the concept of the expected payoff, we first need to consider the terms 'event' and 'exhaustive events': C1 event — Profit, it is equal to tp C2 event — Loss, it is equal to sl P1 — C1 event probability P2 — C2 event probability С1 and С2 events form an exhaustive group of antithetic events i. M4 — expected payoff when closing by a signal.

P1 , P2 — probabilities of stop levels activation provided that one of the stop levels is triggered in any case.

P0[i] — probability of closing a deal with the profit of pr[i] provided that it has not triggered stop levels. i — closing option number P01[j] — probability of closing a deal with the loss of ls[j] provided that it has not triggered stop levels. j — closing option number In other words, we have two antithetic events. PS[k] — probability of setting k th stop level option. MS[k] — expected payoff of closed deals with k th stop levels. M3[k] — expected payoff when closing by a stop order with k th stop levels.

M4 [k] — expected payoff when closing by a signal with k th stop levels. P1 [k] , P2 [k] — probabilities of stop levels activation provided that one of the stop levels is triggered in any case.

P0[i] [k] — probability of closing a deal with pr[i] [k] profit, according to a signal with k th stop levels. i — closing option number P01[j] [k] — probability of closing a deal with ls[j] [k] loss, according to a signal with k th stop levels.

MSp[k] — expected payoff of closed deals with k th stop levels. MSl[k] — expected payoff of closed deals with k th stop levels. M3p[k] — expected payoff when closing by a stop order with k th stop levels. M4p [k] — expected payoff when closing by a signal with k th stop levels. M3l[k] — expected loss when closing by a stop order with k th stop levels. M4l[k] — expected loss when closing by a signal with k th stop levels. For a deeper understanding, I will depict all nested events: In fact, these are the same equations, although the first one lacks the part related to loss, while the second one lacks the part related to profit.

n — total number of steps constant value d — number of steps for price drop u — number of steps for price increase s — total upward movement in steps After defining these values, calculate u and d: To provide the total "s" steps upwards the value can be negative meaning downward steps , a certain number of up and down steps should be provided: "u", "d".

The calculation application screenshot below clarifies this: It lists everything we need. Writing a simple indicator Here I am going to transform my simple mathematical research into an indicator detecting market entry points and serving as a basis for writing EAs.

Let's start from the indicator inputs. To describe the steps, we first need to describe the nodes. It remains to define what and where to call. This will look as follows. Below is the option with other inputs and window style. Review of the most interesting strategies I have developed and seen plenty of strategies. Is it worth the hassle? Does the Grail exist and where to look for it? Tips for common traders All traders want three things: Achieve a positive expected payoff Increase profit in case of a profitable position Reduce loss in case of a losing position The first point is the most important here.

Conclusion Everyone's perspective is subjective. Attached files Download ZIP. zip mq4 mq5 Warning: All rights to these materials are reserved by MetaQuotes Ltd. Copying or reprinting of these materials in whole or in part is prohibited. Other articles by this author Market math: profit, loss and costs The correct way to choose an Expert Advisor from the Market Combinatorics and probability for trading Part V : Curve analysis Combinatorics and probability for trading Part IV : Bernoulli Logic Combinatorics and probability theory for trading Part III : The first mathematical model Combinatorics and probability theory for trading Part II : Universal fractal Combinatorics and probability theory for trading Part I : The basics.

This version implements requested features and provides other improvements, which I found when working with the program. Advanced resampling and selection of CatBoost models by brute-force method This article describes one of the possible approaches to data transformation aimed at improving the generalizability of the model, and also discusses sampling and selection of CatBoost models.

Neural networks made easy Part 3 : Convolutional networks As a continuation of the neural network topic, I propose considering convolutional neural networks. This type of neural network are usually applied to analyzing visual imagery. In this article, we will consider the application of these networks in the financial markets. Timeseries in DoEasy library part 51 : Composite multi-period multi-symbol standard indicators In the article, complete development of objects of multi-period multi-symbol standard indicators.

Using Ichimoku Kinko Hyo standard indicator example, analyze creation of compound custom indicators which have auxiliary drawn buffers for displaying data on the chart. You are missing trading opportunities:. Registration Log in. latin characters without spaces. a password will be sent to this email. Log in With Google. If you do not have an account, please register.

Allow the use of cookies to log in to the MQL5. com website. Various mathematical concepts, statistics, econometrics play a vital role in giving your stock trading that edge in the stock market.

Before starting the mathematical concepts of algorithmic trading , let us understand how imperative is mathematics in trading. In simple words, any individual who buys and sells financial assets in any financial market is a trader.

This individual or trader can trade on the behalf of any other person as well here. A trader is usually someone who trades in shorter time periods as compared to an investor. This simply means that a trader holds assets for a short period to make profits on short-term trends. Whereas, an investor tends to hold assets for a longer-term. A quantitative analyst is the one who designs a complex framework for financial institutions that aids them to price and trade securities in the financial market.

Quants can be of two types:. Moving ahead, now let us find out more about algorithmic trading and its association with Mathematics. Usually, when quants work, they keep an eye on the performance of the market.

But the interesting part is:. Digging deeper, in this process, data is bought from the stock market and is analysed. Those involved in creating algorithms for High-Frequency Trading HFT keep in mind the involvement of a large number of trades in a short period.

For example, in one millisecond the price may go up or go down, and thus, thousands of trades happen in every passing second in HFT. Now, it was not until the late sixties that mathematicians made their first entry into the financial world of Stock Trading. It all started with a professor of mathematics called Edward Thorp, at the University of California, who published a book called Beat the Market in Specifically, Beat the Market concept was nothing but the process of selling the stocks and bonds at one price and then buying them back at a lower price.

It was also observed that in Britain, the fall of the Soviet Union brought an influx of Warsaw Pact scientists. Now let us head to the Mathematical concepts for algorithmic trading which are the core of this article. Starting with the mathematical for stock trading, it is a must to mention that mathematical concepts play an important role in algorithmic trading. Let us take a look at the broad categories of different mathematical concepts here:. Let us walk through descriptive statistics, which summarize a given data set with brief descriptive coefficients.

These can be a representation of either the whole or a sample from the population. Here, Mean, Median and Mode are the basic measures of central tendency. These are quite useful when it comes to taking out average value from a data set consisting of various values.

Let us understand each measure one by one. This one is the most used concept in the various fields concerning mathematics and in simple words, it is the average of the given dataset. Thus, if we take five numbers in a data set, say, 12, 13, 6, 7, 19, 21, the formula of the mean is. Furthermore, the trader tries to initiate the trade on the basis of the mean moving average or moving average crossover. Here, let us understand two types of moving averages based on the ranges number of days of the time period they are calculated in and the moving average crossover:.

A faster moving average is the mean of a data set stock prices calculated over a short period of time, say past 20 days.

A slower moving average is the one that is the mean of a data set stock prices calculated from a longer time period say 50 days. Here, to explain it better, the graph image above is showing three moving lines. Blue one shows the trend line of the stock prices in general.

It is further disintegrated into green and orange lines. The green one indicates a slower- moving average and orange one indicates a faster-moving average. Now starting with the green line, slower moving average the entire trend line shows the varying means of stock prices over longer time periods. The trend line follows a zig-zag pattern and there are different crossovers. For example, there is a crossover between October, and January, where orange line faster-moving average comes from above and crosses the green one slower-moving average while going down.

This indicates that any individual or firm would be selling the stocks at this point since it shows a slump in the market. After the meeting point, ahead both the lines go down and then go up after a point to create one more and then other crossover s. Since there are many crossovers in the graph, you should be able to identify each of them on your own now. On the contrary, it is considered bearish if the faster-moving average drops below the slower-moving average and goes beyond down.

This is so because in the former scenario, it shows that in a short time, there came an upward trend for particular stocks. For example, we will be taking the same instances of the days' moving average for faster-moving average and 50 days' moving average for slower-moving average. Whereas, if the days' moving average goes below the days' moving average, it will be bearish since it means that the stocks fell in the past days.

This period of time can be days, months and even years. Going forward, mean can also be computed with the help of an excel sheet, with the following formula:. Let us understand what we have done in the image above.

The image shows the stock cap of different companies belonging to an industry over a period of time can be days, months, or years. This formula gives the command to the excel to average out the stock prices of all the companies mentioned from row B2 to B6. This is one of the simplest methods to compute Mean. Let us see how to compute the same in python code ahead.

In order to keep it universal, we have taken the daily stock price data of Apple, Inc. from Dec 26, , to Dec 26, You can download historical data from Yahoo Finance. Now, For downloading the Apple closing price data, we will use the following for all python code based calculations ahead:.

Sometimes, the data set values can have a few values which are at the extreme ends, and this might cause the mean of the data set to portray an incorrect picture. Thus, we use the median, which gives the middle value of the sorted data set.

To find the median, you have to arrange the numbers in ascending order and then find the middle value. If the dataset contains an even number of values, you take the mean of the middle two values.

For example, if the list of numbers are: 12, 13, 6, 7, 19, then,. Mainly, the advantage of the median is that unlike the mean, it remains extremely valid in case of extreme values of data set which is the case in stocks.

Median is required in case the average is to be calculated from a large data set, in which, the median shows an average which is a better representation of the data set. Calculation of the median needs the prices to be first placed in ascending order, thus, prices in ascending order are:. The 4th item in the ascending order is INR 75, As you can see, INR 75, is a good representation of the data set, so this will be an ideal one. In the financial world, where market prices vary time and again, the mean may not be able to represent the large values appropriately.

Here, it was possible that the mean value would have not been able to represent the large data set. So, one needs to use the median to find the one value that represents the entire data set appropriately. In the case of Median also, in the image above, we have stock prices of different companies belonging to a particular industry over a period of time can be days, months, or years.

This formula gives the command to the excel to compute the median and as we input the same, we get the result Mode is a very simple concept since it takes into consideration that number in the data set which is repetitive and occurs the most. Also, the mode is known as a modal value, representing the highest count of occurrences in the group of a data.

It is also interesting to note that like mean and median, a mode is a value that represents the whole data set. It is extremely imperative to note that, in some of the cases there is a possibility of there being more than one mode in a given data set. And that data set which has two modes will be known as bimodal. Similar to Mean and Median, Mode can also be calculated in the excel sheet as shown in the image above. SNGL B1: B5. Now, if we take the closing prices prices of Apple from Dec 26, , to Dec 26, , we will find there is no repeating value, and hence the mode of closing prices does not exist.

Coming to the significance of the mode, it is most helpful when you need to take out the repetitive stock price from the previous particular time period. This time period can be days, months and even years. Basically, the mode of the data will help you understand if the same stock price is expected to repeat in the future or not.

Also, the mode is best utilised when you want to plot histograms and visualize the frequency distribution. This brings you to the end of the Measures of Central Tendency. Second, in the list of Descriptive Statistics is Measure of Dispersion. Let us take a look at yet another interesting concept.

It simply tells the variation of each data value from one another, which helps to give a representation of the distribution of the data. Also, it portrays the homogeneity and heterogeneity of the distribution of the observations. This is the most simple out of all the measures of dispersion and is also easy to understand.

Range simply implies the difference between two extreme observations or numbers of the data set. For example, let X max and X min be two extreme observations or numbers. Here, Range will be the difference between the two of them. It is also very important to note that Quant analysts keep a close follow up on ranges. This happens because the ranges determine the entry as well as exit points of trades.

Not only the trades, but Range also helps the traders and investors in keeping a check on trading periods. This makes the investors and traders indulge in Range-bound Trading strategies , which simply imply following a particular trendline. In this, the trader can purchase the security at the lower trendline and sell it at a higher trendline to earn profits.

This is the type which divides a data set into quarters. It consists of First Quartile as Q1, Second Quartile as Q2 and Third Quartile as Q3. The major advantage, as well as the disadvantage of using this formula, is that it uses half of the data to show the dispersion from the mean or average.

You can use this type of measure of dispersion for studying the dispersion of the observations that lie in the middle. This type of measures of dispersion helps you understand dispersion from the observed value and hence, differentiates between the large values in different Quarters.

In the financial world, when you have to study a large data set stock prices in different time periods and want to understand the dispersed value prices from an observed one average-median , Quartile deviation can be used. This type of dispersion is the arithmetic mean of the deviations between the numbers in a given data set from their mean or median average. D0, D1, D2, D3 are the deviations of each value from the average or median or mean in the data set and Dn means the end value in the data set.

These differences or the deviations are shown as D0, D1, D2, and D3, ….. As the mean comes out to be 9, next step is to find the deviation of each data value from the Mean value. So, let us compute the deviations, or let us subtract 9 from each value to find D0, D1, D2, D3, D4, D5, D6, D7, and D8, which gives us the values as such:.

As we are now clear about all the deviations, let us see the mean value and all the deviations in the form of an image to get even more clarity on the same:. Hence, from a large data set, the mean deviation represents the required values from observed data value accurately. It is important to note that Mean deviation helps with a large dataset with various values which is especially the case in the stock market.

Variance is a dispersion measure which suggests the average of differences from the mean, in a similar manner as Mean Deviation does, but here the deviations are squared. Here, taking the values from the example above, we simply square each deviation and then divide the sum of deviated values by the total number in the following manner:.

In simple words, the standard deviation is a calculation of the spread out of numbers in a data set. The symbol sigma represents Standard deviation and the formula is:. Further, in python code, standard deviation can be computed using matplotlib library, as follows:.

All the types of measure of deviation bring out the required value from the observed one in a data set so as to give you the perfect insight into different values of a variable, which can be price, time, etc. It is important to note that Mean absolute data, Variance and Standard Deviation, all help in differentiating the values from average in a given large data set. Visualization helps the analysts to decide on the basis of organized data distribution. There are four such types of Visualization approach, which are:.

Here, in the image above, you can see the histogram with random data on x-axis Age groups and y-axis Frequency. Since it looks at a large data in a summarised manner, it is mainly used for describing a single variable. For an example, x-axis represents Age groups from 0 to and y-axis represents the Frequency of catching up with routine eye check up between different Age groups.

The histogram representation shows that between the age group 40 and 50, frequency of people showing up was highest. Since histogram can be used for only a single variable, let us move on and see how bar chart differs. In the image above, you can see the bar chart. This type of visualization helps you to analyse the variable value over a period of time. For an example, the number of sales in different years of different teams. You can see that the bar chart above shows two years shown as Period 1 and Period 2.

Since this visual representation can take into consideration more than one variable and different periods in time, bar chart is quite helpful while representing a large data with various variables. Above is the image of a Pie chart, and this representation helps you to present the percentage of each variable from the total data set.

Whenever you have a data set in percentage form and you need to present it in a way that it shows different performances of different teams, this is the apt one. For an example, in the Pie chart above, it is clearly visible that Team 2 and Team 4 have similar performance without even having to look at the actual numbers. Both the teams have outperformed the rest.

Also, it shows that Team 1 did better than Team 3. Since it is so visually presentable, a Pie chart helps you in drawing an apt conclusion. With this kind of representation, the relationship between two variables is clearer with the help of both y-axis and x-axis. This type also helps you to find trends between the mentioned variables.

In the Line chart above, there are two trend lines forming the visual representation of 4 different teams in two Periods or two years. Both the trend lines are helping us be clear about the performance of different teams in two years and it is easier to compare the performance of two consecutive years.

It clearly shows that in Period, 1 Team 2 and Team 4 performed well. Whereas, in Period 2, Team 1 outperformed the rest. Okay, as we have a better understanding of Descriptive Statistics, we can move on to other mathematical concepts, their formulas as well as applications in algorithmic trading. Now let us go back in time and recall the example of finding probabilities of a dice roll. This is one finding that we all have studied. Given the numbers on dice i.

Such a probability is known as discrete in which there are a fixed number of results. Now, similarly, probability of rolling a 2 is 1 out 6, probability of rolling a 3 is also 1 out of 6, and so on. A probability distribution is the list of all outcomes of a given event and it works with a limited set of outcomes in the way it is mentioned above. But, in case the outcomes are large, functions are to be used. If the probability is discrete, we call the function a probability mass function.

For discrete probabilities, there are certain cases which are so extensively studied, that their probability distribution has become standardised. We write its probability function as px 1 — p 1 — x. Now, let us look into the Monte Carlo Simulation in understanding how it approaches the possibilities in the future, taking a historical approach.

It is said that the Monte Carlo method is a stochastic one in which there is sampling of random inputs to solve a statistical problem. Well, simply speaking, Monte Carlo simulation believes in obtaining a distribution of results of any statistical problem or data by sampling a large number of inputs over and over again.

Many aim to learn algorithmic trading from the mathematical point of view. Various mathematical concepts, statistics, econometrics play a vital role in giving your stock trading that edge in the stock market. Before starting the mathematical concepts of algorithmic trading , let us understand how imperative is mathematics in trading. In simple words, any individual who buys and sells financial assets in any financial market is a trader. This individual or trader can trade on the behalf of any other person as well here.

A trader is usually someone who trades in shorter time periods as compared to an investor. This simply means that a trader holds assets for a short period to make profits on short-term trends. Whereas, an investor tends to hold assets for a longer-term. A quantitative analyst is the one who designs a complex framework for financial institutions that aids them to price and trade securities in the financial market. Quants can be of two types:. Moving ahead, now let us find out more about algorithmic trading and its association with Mathematics.

Usually, when quants work, they keep an eye on the performance of the market. But the interesting part is:. Digging deeper, in this process, data is bought from the stock market and is analysed. Those involved in creating algorithms for High-Frequency Trading HFT keep in mind the involvement of a large number of trades in a short period.

For example, in one millisecond the price may go up or go down, and thus, thousands of trades happen in every passing second in HFT. Now, it was not until the late sixties that mathematicians made their first entry into the financial world of Stock Trading.

It all started with a professor of mathematics called Edward Thorp, at the University of California, who published a book called Beat the Market in Specifically, Beat the Market concept was nothing but the process of selling the stocks and bonds at one price and then buying them back at a lower price. It was also observed that in Britain, the fall of the Soviet Union brought an influx of Warsaw Pact scientists.

Now let us head to the Mathematical concepts for algorithmic trading which are the core of this article. Starting with the mathematical for stock trading, it is a must to mention that mathematical concepts play an important role in algorithmic trading. Let us take a look at the broad categories of different mathematical concepts here:.

Let us walk through descriptive statistics, which summarize a given data set with brief descriptive coefficients. These can be a representation of either the whole or a sample from the population. Here, Mean, Median and Mode are the basic measures of central tendency. These are quite useful when it comes to taking out average value from a data set consisting of various values.

Let us understand each measure one by one. This one is the most used concept in the various fields concerning mathematics and in simple words, it is the average of the given dataset. Thus, if we take five numbers in a data set, say, 12, 13, 6, 7, 19, 21, the formula of the mean is. Furthermore, the trader tries to initiate the trade on the basis of the mean moving average or moving average crossover.

Here, let us understand two types of moving averages based on the ranges number of days of the time period they are calculated in and the moving average crossover:. A faster moving average is the mean of a data set stock prices calculated over a short period of time, say past 20 days. A slower moving average is the one that is the mean of a data set stock prices calculated from a longer time period say 50 days.

Here, to explain it better, the graph image above is showing three moving lines. Blue one shows the trend line of the stock prices in general. It is further disintegrated into green and orange lines. The green one indicates a slower- moving average and orange one indicates a faster-moving average. Now starting with the green line, slower moving average the entire trend line shows the varying means of stock prices over longer time periods.

The trend line follows a zig-zag pattern and there are different crossovers. For example, there is a crossover between October, and January, where orange line faster-moving average comes from above and crosses the green one slower-moving average while going down. This indicates that any individual or firm would be selling the stocks at this point since it shows a slump in the market. After the meeting point, ahead both the lines go down and then go up after a point to create one more and then other crossover s.

Since there are many crossovers in the graph, you should be able to identify each of them on your own now. On the contrary, it is considered bearish if the faster-moving average drops below the slower-moving average and goes beyond down. This is so because in the former scenario, it shows that in a short time, there came an upward trend for particular stocks. For example, we will be taking the same instances of the days' moving average for faster-moving average and 50 days' moving average for slower-moving average.

Whereas, if the days' moving average goes below the days' moving average, it will be bearish since it means that the stocks fell in the past days.

This period of time can be days, months and even years. Going forward, mean can also be computed with the help of an excel sheet, with the following formula:. Let us understand what we have done in the image above. The image shows the stock cap of different companies belonging to an industry over a period of time can be days, months, or years.

This formula gives the command to the excel to average out the stock prices of all the companies mentioned from row B2 to B6. This is one of the simplest methods to compute Mean. Let us see how to compute the same in python code ahead.

In order to keep it universal, we have taken the daily stock price data of Apple, Inc. from Dec 26, , to Dec 26, You can download historical data from Yahoo Finance. Now, For downloading the Apple closing price data, we will use the following for all python code based calculations ahead:. Sometimes, the data set values can have a few values which are at the extreme ends, and this might cause the mean of the data set to portray an incorrect picture.

Thus, we use the median, which gives the middle value of the sorted data set. To find the median, you have to arrange the numbers in ascending order and then find the middle value. If the dataset contains an even number of values, you take the mean of the middle two values.

For example, if the list of numbers are: 12, 13, 6, 7, 19, then,. Mainly, the advantage of the median is that unlike the mean, it remains extremely valid in case of extreme values of data set which is the case in stocks. Median is required in case the average is to be calculated from a large data set, in which, the median shows an average which is a better representation of the data set. Calculation of the median needs the prices to be first placed in ascending order, thus, prices in ascending order are:.

The 4th item in the ascending order is INR 75, As you can see, INR 75, is a good representation of the data set, so this will be an ideal one. In the financial world, where market prices vary time and again, the mean may not be able to represent the large values appropriately. Here, it was possible that the mean value would have not been able to represent the large data set.

So, one needs to use the median to find the one value that represents the entire data set appropriately. In the case of Median also, in the image above, we have stock prices of different companies belonging to a particular industry over a period of time can be days, months, or years.

This formula gives the command to the excel to compute the median and as we input the same, we get the result Mode is a very simple concept since it takes into consideration that number in the data set which is repetitive and occurs the most. Also, the mode is known as a modal value, representing the highest count of occurrences in the group of a data.

It is also interesting to note that like mean and median, a mode is a value that represents the whole data set. It is extremely imperative to note that, in some of the cases there is a possibility of there being more than one mode in a given data set.

And that data set which has two modes will be known as bimodal. Similar to Mean and Median, Mode can also be calculated in the excel sheet as shown in the image above. SNGL B1: B5. Now, if we take the closing prices prices of Apple from Dec 26, , to Dec 26, , we will find there is no repeating value, and hence the mode of closing prices does not exist. Coming to the significance of the mode, it is most helpful when you need to take out the repetitive stock price from the previous particular time period.

This time period can be days, months and even years. Basically, the mode of the data will help you understand if the same stock price is expected to repeat in the future or not. Also, the mode is best utilised when you want to plot histograms and visualize the frequency distribution. This brings you to the end of the Measures of Central Tendency. Second, in the list of Descriptive Statistics is Measure of Dispersion.

Let us take a look at yet another interesting concept. It simply tells the variation of each data value from one another, which helps to give a representation of the distribution of the data. Also, it portrays the homogeneity and heterogeneity of the distribution of the observations. This is the most simple out of all the measures of dispersion and is also easy to understand. Range simply implies the difference between two extreme observations or numbers of the data set. For example, let X max and X min be two extreme observations or numbers.

Here, Range will be the difference between the two of them. It is also very important to note that Quant analysts keep a close follow up on ranges. This happens because the ranges determine the entry as well as exit points of trades. Not only the trades, but Range also helps the traders and investors in keeping a check on trading periods.

This makes the investors and traders indulge in Range-bound Trading strategies , which simply imply following a particular trendline.