Modelling the alternative support and resistance levels with the application of the linear regression

Modelling the alternative support and resistance levels with the application of the linear regression Wojciech Jakub Podobas Kozminski University Investment Club “Kapitalni” info@tradingalgorithms.pl 1. Abstract The aim of the article is to model a new formula that can calculate price corridor using the linear regression and standard deviation instead of the moving average. The research will check the possible application of the linear regression into the place of the standard moving average. The application of the linear regression instead of moving average and Bollinger Bands could give a better and closer estimated data about price trend, due to the mathematical nature of the linear regression that uses squares from variables instead of the normal arithmetical mean. The application of the linear regression could create a good analytical background for investors to omit jigsaws, bear and bull traps. Research question: How does the linear regression could improve predicting the price movement in the foreign exchange market? 2. Introduction and the background information To understand properly the content of the work few concepts need to be implemented. Key definitions Financial instrument- Every asset that could be traded at the stock exchange1 EUR/USD- currency pair comparing price of Euro [EUR] to American Dollar [USD] 2 Technical analysis- financial mathematics branch defining the price movement at the stock exchange Price trend- the main direction where the price of a financial instrument is going3 Moving average- the trend following technical analysis tool, the continuously changing mean of the price of the financial instrument in the fixed number of time, for example MA20 relates to the average price of the financial instrument in last 20 days4 1 Reuters-Report. (2001). Foreign currency market. New York: Reuters print. p.227 Ibid., p.229 3 Archelis S. (1998). Technical analysis from A to Z. New York: Irwin. p. 72 4 Ibid., p. 15 2 © Kozminski University Investment Club “Kapitalni” 1 Resistance level- the price considered as point where most of the investors starts to selling their positions5 Support level- the price considered as point where most of the investors start to buying new positions6 Price corridor- the price range between the resistance and support levels7 Jigsaw – an entry to the market that results in loses due to false analysis What is the Foreign Exchange Market [FOREX] Foreign Exchange is a non-centralized international market where investors and institutions are exchanging currencies 8 . It was created to allow to exchange one currency to another immediately. Nowadays forex is also a place where people are speculating – buying one currency with hope it will be more vulnerable in future. Forex is inaccurately associated with gambling, because financial markets are only burdened with risk that could be calculated. Base on that, there are many ways to predict future price movements and one of this ways use mathematics. Base theory – the technical analysis Mathematics is being used in predicting price movement of a currency in the form of technical analysis of the stock exchange. The technical analysis assumes that future can be predicted using information and events from the past. The technical analysis examines financial instrument’s charts and behaviour of this charts in past. Technical analysis contains dozens of tools 9 as moving averages from fixed number of period, oscillators or indexes. The linear regression could be applied as one of the tools of technical analysis. To understand how does the linear regression could be applied in the predicting the chart movement it is necessary to realise basics and assumptions of the technical analysis. Principles of the technical analysis The history of price movement could be analysed by the stock bars called candlesticks10. There are two types of candlesticks – increasing and decreasing. Each candlestick denotes a price movement in a given period of time – candlesticks could be construct from seconds, hours to days or even weeks. The timeframe used during the exploration will use candlesticks created for one day. Figure 1. demonstrates notation of these bars. 5 Ibid., p. 17 Ibid., p. 18 7 Ibid., p. 19 8 Reuters-Report. (2001). Foreign currency market. New York: Reuters print. p. 12 9 Chan J. (2011). Technical analysis How to trade like a professional. London: Pearson Education Lt. p. 5 10 Walker W. (2017). Technical analysis for Forex explained. London: Amazon Print. p. 10 6 © Kozminski University Investment Club “Kapitalni” 2 Figure 1 – Candlesticks as a notation of price history (source: own elaboration) There are three possible direction, where price of the financial instrument can move; upwards, downwards and sideways. When price of the financial instrument is moving continuously to the same direction the situation is called a trend. There are also three types of trends; upward trend, downward trend and sideways trend 11 . The very important dependence of price movement is that trend is never going only up or only down – there are always highs and lows. The types of trends are presented at the Figure 2. Figure 2 – The three types of price trend (source: own elaboration)   11 During the uptrend (A) every next highs and lows is higher than previous one. During the downtrend (C) every next highs and lows in lower than previous one. Milewski M. (2011). Forex. Warszawa: SamoSedno. p. 45 © Kozminski University Investment Club “Kapitalni” 3  During the sideways trend (B) there is no rule about highs and lows and price remain in the similar level The reason of this chart behaviour is psychological effect 12 that most of investors are holding their positions only for short period of time and after a satisfactory profit or loss they sell them. The mechanics of the Resistance and Support levels It is easy to see that during a trend price is moving in fixed passage. The price corridor is the range of the financial instrument’s price where investors are feeling confident13 – they think this is real, appropriate price of the financial instrument and they are afraid when price in breaking through. When price is going too low most of investors think that the price of a financial instrument is underestimated and they are lifting price again higher by buying it, when price of the financial instrument is going too high investors think that the price is overestimated and they are selling their positions afraid of reverse in trend. Lower line of the price corridor is called the support level, when higher line of price corridor is called the resistance level. Figure 3. - The resistance and support level in the sideways trend (source: own elaboration) Assuming that price is going in fixed corridor and it is bouncing from the support and the resistance level it is possible to predict price movement and use this information in an investing strategy. For example, an investor could try to buy the currency near to the support level and selling it near to the resistance level. The standard way of calculating resistance and support level is calculation of moving average and stretching it up and down by standard deviation that is named as Bollinger Bands. Moving Average 15 in daily perspective is a mean price of last 15 days middle prices – the middle point between high and low of each day. Moving average is changing every day because every next day is added and the oldest – 16th day is not still counted. 12 13 Chan J. (2011). Technical analysis How to trade like a professional. London: Pearson Education Lt. p. 52 Archelis S. (1998). Technical analysis from A to Z. New York: Irwin.p. 18 © Kozminski University Investment Club “Kapitalni” 4 Possible change in way that support and resistance level is calculated As the aim of the article is to check whether the linear regression could model the price corridor in the better way than the moving average, the linear regression theory needs to be introduced. The work will introduce the method of calculation of the price corridor with linear regression. The results obtained by the new formula will be compared and contrasted with the standard method with Moving Average (Bollinger Bands). Linear regression as a tool in the technical analysis Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable. For example, a modeler might want to relate the weights of individuals to their heights using a linear regression model.14 A linear regression line has an equation of the form Y = ax + b, where x is the explanatory variable and Y is the dependent variable15 In the modelling formulas with linear regression Y contributes to projected price of the pair EUR/USD and x relates to the date. In the foreign exchange the explanatory variable would be the period of time, when the dependent variable will be the middle price during one period. After calculating the linear regression of chosen time (for example one week, one month or even 24 hours) the linear regression formula would give us the exact direction where price was moving and predict where it is going to. 3. Classic methods of calculations for the price corridor by Bollinger Bands The Standard method of calculation of support and resistance level uses moving average. Price corridor for forex pair EUR/USD in May 2017 Calculation of moving average for currency graph shown at the Figure 4. To calculate moving average the middle prices (mean of maximum and minimum day price) of further periods need to be added – every next day will be the next variable in moving average omitting the oldest day. The proper number of periods for moving average to calculate for fixed time is half of periods that are in time that we are interested in 16. Therefore, for one month period, where one period is one day the proper moving average will be for 15 days – MA15.The Figure 5. shows average price for every day in MAY 2017 for EUR/USD pair. 14 http://www.stat.yale.edu/Courses/1997-98/101/linreg.htm (visited 10.08.2017) 15 Ibid. (visited 10.08.2017) 16 Archelis S. (1998). Technical analysis from A to Z. New York: Irwin. p. 72 © Kozminski University Investment Club “Kapitalni” 5 Figure 4. - Graph of price of EUR/USD in May 2017 (source: own elaborated screenshot via IQoptions) Date 01.05.2017 02.05.2017 03.05.2017 04.05.2017 05.05.2018 06.05.2018 07.05.2018 08.05.2018 09.05.2019 10.05.2019 11.05.2019 12.05.2019 13.05.2020 14.05.2020 15.05.2020 16.05.2020 17.05.2021 18.05.2021 19.05.2021 20.05.2021 21.05.2022 22.05.2022 23.05.2022 24.05.2022 25.05.2023 26.05.2023 27.05.2023 28.05.2024 29.05.2024 30.05.2024 Average Price of EUR/USD 1.0910 1.0905 1.0908 1.0905 1.0910 1.0905 1.0933 1.0980 1.0960 1.0892 1.0873 1.08668 1.0896 1.0885 1.095 1.1030 1.1119 1.113 1.116 1.1213 1.1209 1.1214 1.1211 1.1209 1.1211 1.1193 1.1170 1.1171 1.1181 1.1151 Every point of moving average at the graph is calculating by the formula1: MA (for n periods of time) = Where: ∑𝑛 𝑘=1 y 𝑛 𝑛 ∑ y = 𝑦1 + 𝑦2 … + 𝑦𝑛 𝑘=1 MA – moving average for the n periods y – middle price in the one period k – first period The moving average is being calculated automatically by the investing platform and added to the graph (Figure 6.). – Figure 5. – The set of average price of EUR/USD in each day of May 2017 (source: own elaboration via IQoptions) © Kozminski University Investment Club “Kapitalni” 6 Figure 6. - Moving Average 15 of the EUR/USD in May 2017 (source: own elaborated screenshot via IQoptions) Next, to calculate the support and resistance level the MA15 will be moved up and down by the value of Standard Deviation [SD] for each day. Calculating the standard deviation17: 𝑛 |y−y|2 ∑ ̅ SD=√ 𝑘=1 n Where: SD − standard deviation of the price set y −value of average price in one period y̅ − 𝑚𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 n − number of periods k −first period Due to moving average is not constant the standard deviation needs to be calculated for every day – then the moving average will be moved upwards and downwards by the value of SD each day and a resistance and support level will be created continuously. Therefore support and resistance level patter will be changing every day. 17 Chan J. (2011). Technical analysis How to trade like a professional. London: Pearson Education Lt. p. 89 © Kozminski University Investment Club “Kapitalni” 7 Modelling the support and resistance levels patterns: Support level pattern (for one period) = Where: y −value of average price in one period y̅ − 𝑚𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 n − number of periods k −first period ∑𝑛 𝑘=1 y 𝑛 𝑛 𝑛 ∑ − √ 𝑘=1 |y−y ̅|2 n 𝑛 |y−y|2 ∑ y √∑𝑘=1 ̅ Resistance level pattern (for one period)= 𝑘=1 + 𝑛 n Where: y −value of average price in one period y̅ − 𝑚𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 n − number of periods k −first period The support and the resistance levels created using MA15 are shown at the Figure 7. The lines are calculated using computer program (IQoptions), that calculates it more accurately – by calculating it for every 15 minutes interval, therefore the lines are incessant, while my formulas assumes that there is a constant value for each day. Figure 7. - Price corridor of EUR/USD created using EUR/USD in May 2017 (source: own elaborated screenshot via IQoptions) It is easy to see that the resistance and the support level show clearly possible price corridor. The discrepancies between my formula and the computer generated lines are not important, as the mean value of support and resistance level for each level stay the same. © Kozminski University Investment Club “Kapitalni” 8 4. Application of linear regression To calculate the alternative price corridor the equation for resistance and support level needs to be established. Modelling the pattern Using the linear regression it is pointless to use standard deviation for each day, so on one mean standard deviation can be calculated. After that the linear regression will be moved up and down by value of the average standard deviation for whole month. Linear regression pattern: Linear regression pattern=ax +b For a= And 𝑥 ̅) ∑𝑘=1 (x−x̅)(𝑦−y 𝑥 ∑ 𝑘=1 (x−x̅)2 For b=y̅ − 𝑎x̅ Where: x − number of day y −average price for day x x̅ − the mean day of set y̅ − average price of the data set k − first period By moving linear regression line by standard deviation up and down the pattern for the support and resistance level can be written as: Resistance level pattern: ∑ 𝑛 |y−y ̅|2 ax +b + √ 𝑘=1 n Support level pattern: ∑ 𝑛 |y−y ̅|2 ax+b - √ 𝑘=1 n Where: ax +𝑏 − the value of linear regression y −average price for day x y̅ − average price of the data set n – number of periods k − first period Calculating the linear regression function for the EUR/USD price in May 2017. To calculate the price corridor for EUR/USD pair in May 2017 we simply need to substitute values to previously obtained formulas, but to do this it is necessary to calculate the linear regression and standard deviation for whole month that values are shown in the Figure 5. Calculating mean price from the Figure 5: © Kozminski University Investment Club “Kapitalni” 9 ∑𝑛 𝑦 𝑦̅= 𝑘=1 n Where: y −value of average price in one period y̅ − 𝑚𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 n − number of periods k −first period From GDC: 𝑦̅=1.1028 Calculating the Standard deviation: To calculate the alternative price corridor the values of EUR/USD from May 2017 will be substituted to the previous obtained equations. ∑ 30 |y−1.1028|2 SD = √ 𝑘=1 30 Where: y−value of the average price in one period ▲ substituting into values from Figure 5. From GDC: SD= 0.014 By moving linear regression line by standard deviation the support and resistance level will be obtained: Resistance level pattern: 𝑥 ∑𝑘=1 (x −x̅)(𝑦−y̅) ∑ 𝑥 (x − x̅)2 𝑘=1 𝑥 + y̅ − Support level pattern: 𝑥 ∑𝑘=1 (x −x̅)(𝑦−y̅) ∑ 𝑥 (x − x̅)2 𝑘=1 𝑥 + y̅ − 𝑥 ∑𝑘=1 (x −x̅)(𝑦−y̅) ∑ 𝑥 𝑥 (x − x̅)2 𝑘=1 ∑𝑘=1 (x −x̅)(𝑦−y̅) ∑ 𝑥 (x − x̅)2 𝑘=1 x+√ ∑𝑛𝑘=1|y − y̅|2 n x−√ ∑𝑛𝑘=1|y − y̅|2 n Where: x −number of day y −value of average price for day x y̅ − mean price of the data set n − number of periods x̅ − the mean day in the data set k − first period © Kozminski University Investment Club “Kapitalni” 10 ▲ substituting into values from Figure 5. 30 30 ∑𝑘=1 (x −15)(𝑦−1.1028) ∑𝑘=1 (x −15)(𝑦−1.1028) 𝑥 + 1.1028 − x 30 30 2 2 ∑ (x − 15) ∑ (x − 15) 𝑘=1 + ∑30 |y √ 𝑘=1 − 1.1028|2 30 𝑘=1 =0.001356x + 1.0847 + 0.014= 0.001356x + 1.0987 (the value for the resistance level for each day) Hence Support level: 0.001356x + 1.0847 - SD= 0.001356x + 1.0707 The functions for both resistance and support level were calculated and now are added to the graph with the linear regression to estimate the price corridor. Owing to the calculation via IQoptions, the function is calculated more accurately, than for each day, therefore the linear regression will have incessant slope. The Figure 8. shows obtained results of the price corridor. Figure 8. - Price corridor of EUR/USD created using linear regression in May 2017 (source: own elaborated screenshot via IQoptions) © Kozminski University Investment Club “Kapitalni” 11 5. Analysis of both methods of calculation There are two main factors that decide about the analytical values created by the price corridor18: - The number of periods that price is inside the range of corridor The area of the corridor that shows how accurate it is To check which of the two methods is better to predict the price movement and avoid jigsaws both of the mentioned factors will be considered. Comparison and Contrast of two methods of calculating the support and resistance level Figure 9. - Comparison of obtained price corridors of EUR/USD in May 2017 (source: own elaborated screenshot via IQoptions) Analysis and evaluation The graph above shows two price corridors obtained using other technics – moving average 15 and linear regression. Analysing the support and resistance level created using MA15 it is easy to see that they are broken by the price movement during 6th, 15th, 16th and 17th of May. In contrast support and resistance level obtained using linear regression is never broken and the price seems to bouncing from one level to the other. The support and resistance level distance from middle established by MA15 is changing daily what suggest smaller or bigger price movement in other periods. In contrast the support and 18 Archelis S. (1998). Technical analysis from A to Z. New York: Irwin. p. 18 © Kozminski University Investment Club “Kapitalni” 12 resistance level established by linear regression is constant that suggests constant growing value of these levels. When any moving average or linear regression direction in upward this is sign of uptrend. The Moving Average 15 is directed upwards from 1st to 6th and 14th to 30th of May and directed sideways from 7th to 13th of May. The change in direction after 6th of May suggests to investors that situation is not sure and this is better to wait for situation become clearer. The advantage of that is that it is possible to omit losses resulting of fall in EUR/USD price from 6th to 11th May. The linear regression of EUR/USD price in May 2017 is directed upwards that predicts increases in price in future. The linear regression ignore the lows and hills and shows clearly only the general trend. Assume that an investor bought EUR/USD in 5th of May. Unfortunately, hi register loss in next five days. If he will be patient to let the trend works, he would work out losses and register gains in longer period of time. The reason of these result is nature of linear regression. Linear regression can establish a trend more preciously because itis using squares of distances from average prices rather than simply distance as moving average does. Looking at the graph it is good to assume that support and resistance level created using linear regression is a margin, last border for price movement. The bouncing of price from the support level is giving perfect opportunity to open a position. Calculation of the accuracy of both price corridors The best way to see the accuracy of the price corridor is to calculate the area of the track. The smaller the area the more accurate the corridor is, because the price range is tidier – hence it predicts the price movement more accurate. Figure 10. – Areas ticked by both price corridors (source: own elaborated screenshot via IQoptions) © Kozminski University Investment Club “Kapitalni” 13 For the method with linear regression the area can be calculated using simple formula for the area of trapeze. As the price corridor adopts the shape 𝑛 |y−y ̅| ∑ 𝑛+2√ 𝑘=0 n 2 Area= ℎ 2 Where: y −value of average price for day x y̅ − mean price of the data set n − number of periods x̅ − the mean day in the data set k − first period h – the value of two standard deviations between the support and resistance level ▲ substituting into values from Figure 5. From GDC: Area=33.28 The calculation of area for the method with moving average is more complicated. The easiest way to calculate approximated area is the application definite integration. The area between the support and resistance level could be measured, by calculating the definite integral for resistance level and subtract the integral of support level. The area calculated under the bot levels will be approximated, because the integral will be calculated for the mean value of the both level for each day, not for the incessant line as it is shown at the Figure 10. The sum of the integrals from each day will give the area between the line. The area under the resistance level can be calculated by the equation: 𝐴𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑡ℎ𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 = ∫ 𝑛 𝑘 𝑛 ∑𝑛 |y ∑𝑛𝑘=1 y − y̅|2 𝑑𝑥 𝑑𝑥 + ∫ √ 𝑘=1 𝑛 n 𝑘 Where: y −value of average price for day x y̅ − mean price of the data set n − number of periods (last day in the data set) x̅ − the mean day in the data set k − first period (first day is the data set) © Kozminski University Investment Club “Kapitalni” 14 Than the area under the support level needs to be subtracted, and the whole procedure must take place for every one day in the month, therefore the area of price corridor could be written as: 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑒 𝑐𝑜𝑟𝑟𝑖𝑑𝑜𝑟: 𝑛 𝑛 ∑𝑛 y 𝑑𝑥 ∑ ∫𝑘 𝑘=1 𝑛 𝑘=0 𝑛 ∑ 𝑘=0 +∑ 𝑛 ∫𝑘 √ 𝑛 ̅|2 ∑𝑘=1|y−y n 𝑛 𝑘=0 𝑑𝑥 ) 𝑛 |y−y ̅|2 𝑛 ∑ ∫𝑘 √ 𝑘=1n 𝑑𝑥 − ( ∑ 𝑛 𝑘=0 𝑛 ∑𝑛 𝑘=1 y ∫𝑘 𝑛 𝑑𝑥 − Where: y −value of average price for day x y̅ − mean price of the data set n − number of periods (last day in the data set) x̅ − the mean day in the data set k − first period (first day in the data set) ▲ substituting into values from Figure 5. Calculating atomically using Wolfram Alpha Calculator (visited 01.12.2017 https://www.wolframalpha.com): Area = ~39.56 Comparing the two values – 33.28 and 39.56 we can say that the price corridor that uses the linear regression is more accurate because it is tracking smaller area and, therefore is providing more precise price range for the trader. 6. Conclusion and Further Evaluation It is impossible to state which one method is absolutely better. Both methods consist advantages and disadvantages. The good point of information is that linear regression can be better for an investor that would like to buy EUR/USD for longer period of time, when MA15 will be better for short-term investors. It is also proven that the method with linear regression is more accurate and consists smaller error overall. It is very important for the trader to avoid jigsaws. The application of modelled formulas for resistance and support level using linear regression is giving a new future-proof technical analysis tool that could be combined with any investing strategy. The observation that price corridor calculated using linear regression shows other features of price graph than standard price corridor is very important for the technical analysis because it could eliminate some mispredictions based only on moving average application. In conclusion, obtained patterns for trend line and support and resistance level with an application of linear regression can affect many trading strategies – including mine. Many investors and traders could be better off by studying the results of alternative price corridor. © Kozminski University Investment Club “Kapitalni” 15 Extension and an improvement of the paper The mathematical exploration shows alternative method of calculation of price corridor for any financial instrument. The method is flexible and established support and resistance levels patterns could be used for more time frames. The major weakness of the experiment is that the alternative price corridor was established only for one month for one currency, that create many doubts about the method, as the positive outcome could be caused only by fortune. What is more, the formulas for calculation the support and resistance level using the moving average were not totally accurate, as the resistance and support level are normally calculated for every second, not only for one day as during the exploration. Therefore the integrals obtained using the self-created mathematical formulas were approximate, and the area of the price corridor created by moving averages may differ a bit. The possible improvement for the experiment is research for more data sets as more timeframes of the EUR/USD pair, more assets as stocks or commodities. While the prediction of the linear regression application was proven and the price corridor established via it works, the research sample is too small to state if it will work for any time frame and any currency pair as for example for people trading hourly at the pair GBP/JPY. The best improvement for the research would be setting price corridors for bigger number of sets in many timeframes, not only for monthly frame for the pair EUR/USD. What is more, to calculate the actual area of the price corridor established using moving averages requires more computing power and more sophisticated mathematical formulas that ought to be implemented to be clearly certain about the new method. 7. Bibliography Archelis S. (1998). Technical analysis from A to Z. New York: Irwin. Chan J. (2011). Technical analysis How to trade like a professional. London: Pearson Education Lt. Walker W. (2017). Technical analysis for Forex explained. London: Amazon Print. Milewski M. (2011). Forex. Warszawa: SamoSedno. Reuters-Report. (2001). Foreign currency market. New York: Reuters print. Internet resources: http://www.stat.yale.edu/Courses/1997-98/101/linreg.htm (visited 10.08.2017) https://www.wolframalpha.com/ (visited 01.12.2017) © Kozminski University Investment Club “Kapitalni” 16