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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. 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.
Also, it says that this way we can outperform the market without any risk. One example of Monte Carlo simulation can be rolling a dice several million times to get the representative distribution of results or possible outcomes. With so many possible outcomes, it would be nearly impossible to go wrong with the prediction of actual outcome in future.
Ideally, these tests are to be run efficiently and quickly which is what validates Monte Carlo simulation. Although asset prices do not work by rolling a dice, they also resemble a random walk. Let us learn about Random walk now. Random walk suggests that the changes in stock prices have the same distribution and are independent of each other. Hence, based on the past trend of a stock price, future price can not be predicted. Also, it believes that it is impossible to outperform the market without bearing some amount of risk.
Coming back to Monte Carlo simulation, it validates its own theory by considering a wide range of possibilities and on the assumption that it helps reduce uncertainty. Monte Carlo says that the problem is when only one roll of dice or a probable outcome or a few more are taken into consideration. Hence, the solution is to compare multiple future possibilities and customize the model of assets and portfolios accordingly. For example, say a particular age group between had recorded maximum arthritis cases in months of December and January last year and last to last year also.
Then it will be assumed that this year as well in the same months, the same age group may be diagnosed with arthritis. This can be applied in probability theory, wherein, based on the past occurrences with regard to stock prices, the future ones can be predicted. There is yet another one of the most important concepts of Mathematics, known as Linear Algebra which now we will learn about. The most important thing to note here is that the Linear algebra is the mathematics of data, wherein, Matrices and Vectors are the core of data.
A matrix or the matrices are an accumulation of numbers arranged in a particular number of rows and columns. Numbers included in a matrix can be real or complex numbers or both. In simple words, Vector is that concept of linear algebra that has both, a direction and a magnitude. In this arrow, the point of the arrowhead shows the direction and the length of the same is magnitude.
Above examples must have given you a fair idea about linear algebra being all about linear combinations. These combinations make use of columns of numbers called vectors and arrays of numbers known as matrices, which concludes in creating new columns as well as arrays of numbers.
There is a known involvement of linear algebra in making algorithms or in computations. Hence, linear algebra has been optimized to meet the requirements of programming languages. This helps the programmers to adapt to the specific nature of the computer system, like cache size, number of cores and so on. Coming to Linear Regression, it is yet another topic that helps in creating algorithms and is a model which was originally developed in statistics. Linear Regression is an approach for modelling the relationship between a scalar dependent variable y and one or more explanatory variables or independent variables denoted x.
Nevertheless, despite it being a statistical model, it helps with the machine learning algorithm by showing the relationship between input and output numerical variables. Machine learning implies an initial manual intervention for feeding the machine with programs for performing tasks followed by an automatic situation based improvement that the system itself works on.
It is such a concept that is quite helpful when it comes to computational statistics. Computational statistics is the interface between computer science and mathematical statistics. Hence, computational statistics, which is also called predictive analysis, makes the analysis of current and historical events to predict the future with which trading algorithms can be created.
In short, Machine learning with its systematic approach to predict future events helps create algorithms for successful automated trading. If you wish to read more on Linear regression and its advanced equations, refer to the link here. In the graph above, x-axis and y-axis both show variables x and y.
Since more sales of handsets or demand x-axis of handsets is provoking a rise in supply y-axis of the same, the steep line is formed. In linear regression, the number of input values x are combined to produce the predicted output values y for that set of input values. Basically, both the input values and output values are numeric. To read more, please refer to the blog here. As we move ahead, let us take a look at another concept called Calculus which is also imperative for algorithmic trading.
Calculus is one of the main concepts in algorithmic trading and was actually termed as infinitesimal calculus , which means the study of values that are really small to be even measured. In general, Calculus is a study of continuous change and hence, very important for stock markets as they keep undergoing frequent changes.
Now, if time t is 1 second and distance covered is to be calculated in this time period which is 1 second, then,. But, if you want to find the speed at which 1 second was covered current speed , then you will be needing a change in time, which will be t. Since t is considered to be a smaller value than 1 second, and the speed is to be calculated at less than a second current speed , the value of t will be close to zero.
This study of continuous change can be appropriately used with linear algebra and also, can be utilised in probability theory. In linear algebra, it can be used to find the linear approximation for a set of values and in probability theory, it can determine the possibility of a continuous random variable. Being a part of normal distribution, calculus can be used for finding out normal distribution as well.
To read more on normal distribution, read here. In the entire article, we have covered various topics on mathematics and statistics in stock trading, that is stock market math, and also the related subtopics of them all. Since algorithmic trading requires a thorough knowledge of mathematical concepts, we have learnt various necessary concepts namely :.
Explaining them all, there are subtopics providing you with important and deeper aspects of each with their mathematical equations and computation on platforms like excel and python. As the entire article is aimed to get you closer to your next step in algorithmic trading. You can join EPAT algorithmic trading course by QuantInsti and learn algorithmic trading in a structured manner from the leading industry experts in online classroom lectures.
Get in touch with programme counsellors today. Disclaimer: All data and information provided in this article are for informational purposes only. All information is provided on an as-is basis. What is the need of learning Math for stock markets? Where do I learn about the application of math in the stock markets?
What are the basics of stock market math? Here's a complete list of everything that are covering about Stock Market ath: Who is a Trader? Who is a Quant or Quantitative Analyst? Why does Algorithmic Trading require Math? What are matrices? What are the vectors? Linear Regression How is Machine Learning helpful in creating algorithms?
Calculating Linear Regression Calculus Before starting the mathematical concepts of algorithmic trading , let us understand how imperative is mathematics in trading. Who is a Trader? Quants can be of two types: Front office quants - These are the ones who directly provide the trader with the price of the financial securities or the trading tools. Back office quants - These quants are there to validate the framework and create new strategies after conducting thorough research.
When and How Mathematics made it to Trading: A historical tour Now, it was not until the late sixties that mathematicians made their first entry into the financial world of Stock Trading. In this book, he claimed that he had provided the foolproof way of earning money on the stock market. This hedge fund proceeded to rule over the markets and hence, it became a full-fledged strategy.
Soon after, a generation of physicists entered the depressed job market. On observing the quantum of money that could be made on Wall Street, many of them moved into finance consequently. This brought along a new concept of quantitative analysis and a mathematics genius named Jim Simons became famous in bringing enough knowledge in the particular sphere. In , Jim Simons also founded an exceptional hedge fund management company called Renaissance Technologies.
Mathematical Concepts for Stock Markets 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: Descriptive Statistics Probability Theory Linear Algebra Linear Regression Calculus Descriptive Statistics Let us walk through descriptive statistics, which summarize a given data set with brief descriptive coefficients.
Mean 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. 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: Faster moving average Shorter time period - A faster moving average is the mean of a data set stock prices calculated over a short period of time, say past 20 days.
Slower moving average Longer time period - 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. In other words, this is when the shorter period moving average line crosses a longer period moving average line. Whereas, in the latter scenario it shows that in the past few days there was a downward trend.
Median 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. Here, the 3rd value in the list is So, the median becomes 12 here. For example, in case the data set is given as follows with values in INR: 75,, 82,, 60,, 50,, 1,00,, 70, and 90, Let us learn how to compute in the python code. Now as you have got a fair idea about Mean and Median, let us move to another method now.
Mode Mode is a very simple concept since it takes into consideration that number in the data set which is repetitive and occurs the most. SNGL B1: B5 B1: B5 - represents the values from cell B1 till 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. In short, it simply shows how much the entire data varies from their average value. Range This is the most simple out of all the measures of dispersion and is also easy to understand.
Of course, when the novices learn about this, they also try to apply their memories of the higher mathematics course in trading. In result, various mathematical Forex strategies appear. Unfortunately, these solutions lead to a waste of time at best, or large sums at worst. Therefore, since the novices are interested in this topic from generation to generation, today we will look at the advantages and disadvantages of the Martingale as a system.
First of all, we should recall that this technique came to the financial markets from the casinos, or rather from the gambling houses. The exact date of its real practical application is not known, but at rough round evaluation, it appeared at the junction of the 18th and 19th centuries, and has gained wide popularity in the 20th century. In the general case, this is such a strategy in gambling that doubles up after each losing bet until it receives a prize.
Thus, the player assumes unlimited risk, but can only win an amount equivalent to the initial bet in the series. Important nuances that need to be taken into account when studying the mathematical Forex strategies. Judging by the polls on independent forums, the deposit siphon off is often a consequence of the use of the Martingale. To answer the question of why such an approach is unacceptable, we should again turn to history.
Initially, the roulette game had only two fields: black and red; accordingly, from the point of view of the theory of probability, the outcomes of tossing a coin and bets in the casino were completely identical, i. The result of this game at a constant bet was the following graph:.
It would seem that it is the grail, because theoretically, even without increasing the bet, it can bring a profit. Since then, the history of the evolution of the Martingale began, because it has become impossible to stay afloat without an increase in bets. Thus, even if we ignore the zero, in the above example 7 consecutive losses were recorded at one sector, which means that if you double the bet after every failure, we get the following sequence of losses: 1, 2, 4, 8, 16, 32, In fact, the funds will not be enough even for the opening of the last series.
Where the Martingale shows better results — in the casino or on Forex? Note that the mathematical Forex strategies are subject to these factors at a greater degree — in particular, the role of zero that reduces the probability of winning in the Forex market is played by spread and DC commission, and they are present in every single transaction, regardless of whether it is profitable or unprofitable. The figure below shows an example of a random process without doubling transactions, the result of experiments of which is adjusted for the potential loss on the commission:.
In addition, in most cases, losing trades are countertrend, so a further increase of the lot against the prevailing trend only exacerbates the situation. This situation is a consequence of the above-mentioned autocorrelation, when every new values of the row depend on the preceding, which creates a lot of problems when searching for repetitive signals.