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Task №6. Basic technical indicators: mathematical analysis

This time we will discuss a wide range of basic tools related to mathematical analysis. In the context of forex trading, mathematical analysis involves a variety of methods and techniques in a trader’s arsenal aimed at forecasting future price movements in the financial market. Most of these methods imply the use of various technical indicators. In other words, this type of analysis is called mathematical because every single technical indicator is based upon strict mathematical calculations.

Today, there is an enormous variety of graphical indicators allowing a trader to predict the direction and strength of market trends. The most common and frequently used mathematical indicators include moving averages and stochastic oscillators.

For a comprehensive study of mathematical analysis as applied to the forex market, one should consider both sides of the subject.

On the one side, mathematical analysis on Forex contains many elements of technical analysis referred to as so-called expert advisors (i.e. automated software used in a trading platform). This is why mathematical analysis forms the cornerstone of numerous trading strategies, being an indispensable part of expert advisors. For example, such well-known strategies as the Martingale betting system, the Puria method, the identification of supply and demand zones in the market are all based on the laws of mathematics.

On the other side, computer data processing is considered to be another important aspect of mathematical analysis. Computing technologies along with the probability theory and statistical data provide the basis for developing cutting-edge trading solutions that have become widespread in the financial world over the recent years. Among computer-based technical indicators are moving averages, Bollinger Bands, stochastic oscillators etc.

Generally speaking, market analysis based on such an exact science as mathematics is an integral component of rational capital management and efficient risk control.

Before we take a closer look at trading indicators, let us introduce the notion of divergence. The divergence is rightfully considered to be the strongest signal in technical analysis. It occurs when the actual price movement and a technical indicator are heading in opposite directions.

A bearish divergence takes place when prices are on the rise, approaching a new high, but the indicator stops at a lower level. Bearish divergences between prices and indicators point to some weakness at the top of the market and signal a higher probability of an upcoming trend reversal.

Bearish divergence

Conversely, if prices are moving towards a new low while the indicator ticks higher, we observe a bullish divergence on the chart.

A bullish divergence between the price and a technical indicator is a clear sign indicating weakness in the trend and a potential shift to the upside.

Bullish divergence

All technical indicators can be roughly divided into two large groups: trend-following indicators and stochastic oscillators.

Trend-following indicators

Before we proceed to specific types of trend-following indicators, let us make sure that you fully understand what a trend is. The traditional definition states that a market trend is a sustainable price movement in a certain direction. There are three types of trends: bullish (uptrend – prices are rising), bearish (downtrend – prices are falling), and sideways (horizontal – prices are stuck in a narrow trading range, hovering up and down with no clear direction). Therefore, trend-following indicators are designed to determine the current market trend and predict changes in the direction of price movement. As a rule, trend-following indicators are highly accurate in locating the point where a trend starts, ends, or reverses. Nowadays, hardly any trader in the currency market can do without these helpful technical tools. In fact, most market participants commonly use more than one trading indicator, seeking to achieve higher accuracy.

Trend-following indicators start identifying a trend after they have been added to the price chart. Remember that the process usually involves a certain time lag. Still, using this type of indicators makes it possible to avoid common errors and false signals that can undermine a trader’s success.

At present there exists a tremendous amount of trading indicators designed specifically for the currency market. Many of them are already available in most trading platforms, which means that these tools enjoy the greatest popularity among market participants. However, there are also some indicators that are not that widely recognized and are only used by traders who are well familiar with their functions and operating principles. Some traders also create their own unique expert advisors to suit their personal needs and trading style. Such customized indicators sometimes gain popularity on a wider scale and become widespread.

To achieve efficient and profitable trading based on trend-following indicators, a trader does not necessarily have to test as many technical systems as possible. The use of two or three expert advisors combined with sufficient knowledge and trading skills would be enough to bring positive results.

Below we will describe several basic trend-following indicators in more detail.

The moving average (MA) is the most frequently used indicator in technical analysis. Moving average lines are added directly to the chart reflecting price movements. A moving average is calculated over a certain time period selected by a trader. The shorter this period is, the higher the probability of getting false signals is. At the same time, longer periods imply a weaker sensitivity of the moving average.

There are different types of moving averages such as

  1. the simple moving average
  2. the weighted moving average
  3. the exponential moving average

Moving averages belong to the category of analytical tools following the trend. They are intended to signal a potential trend reversal and spot the beginning of a new trend and the end of an existing one. Therefore, MAs track the process of trend development. They can be considered curved trend lines. However, a moving average is unable to forecast price shifts in the same way as graphical analysis can predict market dynamics. Instead of running ahead of price movements, it follows behind their dynamics over time. MAs can only indicate the formation of a new trend after (not before) it actually emerged.

Building a moving average is an effective technique for spotting and measuring trends by smoothing price values. Indeed, after higher and lower price deviations have been smoothed out to average values thus filtering out the noise from random fluctuations and flattening the curve, it is much easier for a chartist to monitor the development of present market trends.

A short-term moving average provides more accurate results in tracking price movements than long-term MAs. Time lags can be reduced through the use of shorter time frames but still cannot be eliminated completely. Short-term moving averages work best in a sideways market. Longer-term MAs are preferred in a trending (bullish or bearish) market as they are less sensitive.

Simple moving average

A simple (arithmetic) moving average (SMA) is calculated as follows:

where Рi is the closing prices of day i, and n is the period.

This type of moving averages is widely used by most technical analysts. However, some chartists are skeptical about it, citing two main reasons to doubt the efficiency of SMAs.

First, SMA analysis only takes into account the data contained within the period covered by an SMA.
Second, a simple moving average assigns equal weight to each day’s price values. For example, in a 10-day moving average the price from the first day (10 days ago) carries the same weight (10%) as the price from the last day (yesterday), as well as all the other days’ prices. Similarly, a 5-day moving average implies that the average weight of each day’s price equals 20%.
Meanwhile, some experts argue that more weight should be attributed to more recent price data. This probably makes sense because it takes longer for a simple moving average to signal a reversal whenever a new trend emerges than in the case of the weighted moving average, which we will discuss below.

Weighted moving average

Some technicians use weighted moving averages (WMA) to solve the problems related to average day values. The following formula is used to calculate the WMA:

where Wi is the weight of component i (price).

The weight assigned to prices in the formula above can be selected by a trader depending on the price dynamic of the asset being traded. There are several techniques for smoothing out a series of prices including linear and exponential weighted moving averages. For example, in the case of a linear weighted moving average, Wi (weight) equals i (price).

Exponential moving average

The exponential moving average (EMA) has a more complex structure than the WMA or SMA, making it harder to calculate. It allows a chartist to fix the two weaknesses of a simple moving average.
First, the EMA assigns more significance to recent days’ prices. That is why it is also called exponentially weighted. Even though previous price dynamics has smaller weight, all of the price data is used to calculate an exponential moving average. In this case, the formula looks somewhat more complicated:

EMAt = EMAt-1 + (k * (Pt - EMAt-1)),

where t is the present day, t-1 is the previous day, k equals 2 divided by (n + 1), and n is the period.

Even though the exponential moving average does not have the drawbacks of a simple MA, it is not the most efficient tool out of all the three types we have described. Below we will discuss the effectiveness of each MA type.

Basic rules for analysis:
  1. The most important signal that reflects the direction of a trend is the general direction of an MA movement. With a rising moving average, a chartist would take a bullish approach, attempting to profit from a rise. Buy when prices edge down to the MA, placing a stop order below the previous low and moving it as soon as the price closes below the earlier level. A falling MA means that prices are heading lower in a bearish market. In this case, open short positions whenever prices rise to the MA or anywhere higher. Place a stop order slightly above the previous peak and move it dynamically if the downward trend remains in place.
  2. The second signal to take into account is the crossover of a moving average and the price. This is the strongest signal indicating that a new bullish trend is emerging if an MA crosses the price chart from top to bottom with an upward slope, and the price chart also has a large upward slope. Another type of crossover occurs when a moving average with a downward slope crosses through the price sloping downwards or upwards. Such a crossover gives a weaker signal for an upcoming bullish trend. In this case, you need an additional signal to confirm further market dynamics. Similar signals with opposite slopes should be interpreted as signs of an upcoming bearish market.
  3. The third signal is a reversal of an MA at the highest or lowest price. If a moving average is located below the price chart and has an intraday low while the price chart has an upward slope, it sends a moderate buy signal indicating the presence of a bullish trend. If the price chart does not have an upward slope, a moving average sends a very weak signal. It takes three additional signals to confirm it.

These rules can be applied to trending markets. In a trendless market, such lines will be curved. A trader’s attempts to smooth out these curves manually usually lead to a loss of a useful signal.

You can see three types of moving averages on the chart. The crossover of the market price and the indicator at the bottom gives a buy signal.


In a broad sense, an oscillator is anything that moves back and forth (oscillates) from one extreme (maximum or minimum, high or low) to another, and back. It always operates in one of two states, i.e. either on or off.
On Forex, oscillators represent technical tools that measure the difference between two specific points on a price chart. An oscillator appears as a wavy line that moves lower and higher between two extremes. This indicator serves as a signal to buy or sell an asset. It is also used to determine when a currency pair has reached an overbought or oversold condition. Forex oscillators can range on a scale from 0 to 100. Analysis thus depends on these scale readings. The market is interpreted as being in an overbought area when an oscillator rises above 70 or 80. Values below 20 or 30 denote an oversold condition. Moreover, some oscillators also have a middle (zero) line separating the upper and lower halves of the range. The crossing of the middle line is considered to constitute a buy or sell signal indicating an upcoming trend reversal.
There are various types of oscillators applied in technical analysis. We recommend using those that allow a trader to assess the situation and take the proper decision quickly. Among the most common and frequently used oscillator tools are the Stochastic Oscillator, Relative Strength Index (RSI), Percent Range (%R), Moving Average Convergence/Divergence Oscillator (MACD), Commodity Channel Index (CCI), and Momentum. Stochastic oscillators compare the most recent closing price to the price range between its high and low over a certain time period. The RSI identifies trend strength and indicates potential reversals. The Williams’ Percent Range, developed by famous commodity trader Larry Williams, shows when a currency pair is overbought or oversold depending on the current price as compared to its range over the previous periods. The MACD is based on the analysis of moving averages and serves to determine a trend’s direction. The CCI was initially intended for identifying trend reversal points on the commodity market but later became popular on Forex, too. The Momentum oscillator, invented by technical analyst Tushar Chande, measures the speed of price change.
Keep in mind that oscillators are not infallible as they may sometimes give false signals. Therefore, a trader should use caution when applying this type of technical indicators. Beginners are not advised to start their charting experience with oscillators.

MACD (Moving Average Convergence/Divergence)

The Moving Average Convergence/Divergence (MACD) is a dynamic trend-following indicator reflecting the relationship between two moving averages of prices. The MACD indicator measures the difference between two exponential moving averages (EMA) with default 12- and 26-day periods. To identify the best opportunities for buying or selling an asset, a so-called signal line is plotted on the MACD chart representing a default 9-period exponential moving average.

To calculate the histogram (the divergence between the MACD and the signal line), subtract the value of a 26-period exponential moving average from a 12-period EMA. Then subtract a 9-period EMA of the MACD (the signal line) from the result.

MACD = EMA(P,12) – EMA(P,26) – EMA[(EMA(P,12) – EMA(P,26))9]

where P is the price, and EMA(P, n) is an n-day EMA.

The efficiency of this method has made the MACD indicator extremely popular and widely-used among technical analysts.
The MACD demonstrates the most reliable results on a time scale of one day or more. Be careful when applying MACDs on charts shorter than one day. As for smaller time frames, the performance of MACDs on less-than-an-hour charts may involve a lot of false signals.

MACDs prove more effective when the market fluctuates in a wide trading range. The most popular MACD signals include crossovers, overbought/oversold conditions, and divergences.

Below we will describe three possible applications of MACDs.

1. Trading with MACD crossovers

Buy signals are generated when the histogram crosses the zero line from below, and vice versa for sell signals. Be aware that in most cases mechanical trading with every MACD crossover results in frequent whipsaws and significant losses. The best idea would be to avoid narrow trading ranges that undermine the effectiveness of the indicator. Open a long position when a buy order is generated around the zero line. In most cases, such a signal is preceded by a strong bullish trend. If the signal is generated well below the zero line, it means that the trend will be extremely weak. You can also obtain additional information when analyzing an MACD histogram. If a price bar is well above the zero level, and each consecutive bar is smaller than the previous one, momentum becomes weaker. When displaying the Momentum indicator, the histogram gives earlier signals.

2. Divergence

Divergences represent the second way to use MACDs. The divergence is a very effective pattern for most types of technical analysis, and the MACD is no exception in this case. A divergence occurs when prices and MACD lines are moving in opposite direction. Divergences are most helpful in identifying a trend continuation rather than a potential trend reversal. In any case, it is best not to hurry but to wait for a full divergence to establish on the chart before entering the market. Otherwise you will find yourself on the opposite side of a strong trend.

3. Overbought / Oversold levels

One more way to use the MACD is to identify whether the market is overbought or oversold (or neither). This tool allows a trader to determine the turning points where the market is likely to reverse with a certain degree of probability. If the MACD reaches extreme highs or lows on the opposite sides of the zero line, it means that the market is in an overbought or oversold condition. Considering this type of signals, one needs to determine these extreme points empirically as they would be different for each trading instrument. When the histogram reaches these critical areas, any crossing with the signal line generates a buy/sell signal. Crossovers that occur before the price reaches extreme levels may be ignored, thus eliminating most whipsaws. Based on a small research, such levels can be identified in any market that is in a broad trading range with wide price fluctuations. Signals that occur in the middle area of the MACD chart should be taken into account only if another trusted indicator confirms that the price will be moving in the direction of the trend. Of course, this may be determined by an oscillator. Remember that the MACD works best as a long-term trend-following tool rather than a short-term market timer for trading purposes.

The bullish divergence on the MACD chart warns of a possible change in the trend direction. The break of the zero line gives a buy signal.

ADX (Average Directional Movement Index).

The Average Directional (Movement) Index (ADX), also known as the Directional Movement Index (DMI) was introduced in the mid-1970s by Welles Wilder, who developed a trend-following technique that he called the Directional System. This method was elaborated and modified by a number of technical analysts. The Directional System is used to measure the strength of market trends, i.e. the trendiness of a market. When calculating the ADX, a chartist applies two more indicators: Plus (or Positive) Directional Indicator (+DI) and Minus (or Negative) Directional Indicator (-DI). It takes several steps to build an ADX:

  1. Determine directional movement (DM) by comparing yesterday’s and today’s price range, or the distance from the highest to the lowest levels. Directional movement is the largest part of today’s range that lies outside of yesterday’s range.
  2. Measure the true range (TR) of the market. TR is always a positive value, the largest of the three:
    1. the distance between today’s high and today’s low;
    2. the distance between today’s high and yesterday’s closing price;
    3. the distance between today’s low and yesterday’s closing price.
  3. Calculate daily directional indicators: the Plus Directional Indicator (+DI) and the Minus Directional Indicator (-DI). They are used to compare various markets, expressing their directional movement as a percentage of each market’s true range. Each of the DIs is a positive figure: +DI equals zero if there is no clear uptrend within the day; -DI equals zero if there is no clear downtrend within the day.

  4. Identify smoothed (averaged) directional lines. Smoothed +DI and –DI are created with the help of a 13-day EMA. As a result, there are two smoothed indicator lines, the one is positive and the other is negative. The relationship between positive and negative lines reflects market trends. Once you have completed these actions, you can calculate the Average Directional Movement Index. This component of the Directional System shows when a trader should follow a trend. The ADX measures the difference between two directional lines (+DI and –DI). This index is calculated as follows:

This indicator measures the strength of a trend. A rising ADX implies that a market trend is gaining strength, marking the best moment to place trend-following orders. A falling ADX means that the trend strength is weakening. In this case, a trader is recommended to track signals sent by oscillators.
Directional analysis serves to trace changes in mass optimism and pessimism, measuring the ability of bulls and bears to push prices beyond the limits of the previous day’s range. If today’s high is above yesterday’s high, the market becomes more optimistic. Similarly, if today’s low is below yesterday’s low, then the market sentiment shifts to pessimism.

When analyzing directional indicators, pay attention to these signals:

  1. If the +DI line is above the –DI line, the trend is moving upwards.
    If the +DI line is below the –DI line, the trend is moving downwards.
  2. If the two lines diverge, the directional movement of a trend increases.
    The convergence of the two lines indicates a pause in the existing trend or an upcoming reversal.

Follow these trading rules if you use the Directional system method:

  1. Buy when the +DI rises above the –DI and sell when the -DI is higher than +DI. The best moment to buy is when both the +DI and ADX are above the –DI, with a rising ADX. This setup signals a stronger bullish trend. It is best to sell when the –DI and ADX are above the +DI, with a rising ADX.
  2. A falling ADX indicates that the trend strength is decreasing and further price direction is unclear. When the ADX moves lower or falls below both DI lines, we observe a trendless market. In this case, the use of directional indicators is not recommended.

The best signal in the Directional System occurs when the ADX first falls below both DI lines and then rises. If that is the case, be prepared for a new bullish or bearish trend to emerge. If the ADX rises by four points, for example, from 18 to 22 from its lowest point below both DIs, this is a clear signal that a new trend is about to start. Enter long positions when the +DI is higher and place stop orders below the most recent bottom. Conversely, sell if the –DI is higher.

The rising ADX confirms the continuation of the bullish trend, indicated by the +DI line moving higher, and sends a buy signal.

Bollinger Bands

Bollinger bands are similar to moving average envelopes. The difference between them is that the borders of envelopes are drawn above and below the moving average curve at a fixed distance expressed as a percentage, whereas the boundaries of Bollinger bands are plotted at levels equal to a certain amount of standard deviations. Since the standard deviation value depends on volatility, Bollinger bands are self-adjusting: widening when the market is volatile and narrowing during more stable periods. As a rule, these patterns are built on a price chart but can also be displayed on an indicator chart. Below we will focus on the bands that are plotted on price charts. Similar to the case with moving average envelopes, the interpretation of Bollinger bands is based on the fact that prices tend to stay within the upper and lower boundaries of a band. The distinctive feature of Bollinger bands is their varying width that depends on shifts in price volatility. During the periods of considerable price changes, the bands widen to let prices fluctuate more freely. During the periods of stagnation, the bands narrow thus curbing price volatility and containing prices within their boundaries.

  1. As a rule, sharp price swings occur after the bands narrow due to decreased volatility.
  2. When prices move outside the bands, a trader should expect a continuation of the existing trend.
  3. Peaks and bottoms outside of Bollinger bands followed by peaks and bottoms inside them indicate a potential trend reversal.
  4. Price movement that originated near one of the bands normally reaches the opposite boundary.
  5. This observation is useful for predicting price targets.

Momentum и ROC

The Momentum oscillator measures the price change over a certain time period. It is calculated according to the following formula:

MOM = C – Cn

Rate of Change (ROC) is the second simplest application of oscillator techniques. As well as the Momentum indicator, it displays the difference between today’s price and the price x-time periods ago. There is only one significant difference between the ROC and Momentum as the latter is expressed as a ratio rather than as a differential. The Rate of Change formula looks as follows:

where Рt is today’s closing price.

Рх is the closing price x days ago.

The Momentum and Rate of Change look almost identical when plotted on a price chart and are applied in a similar manner. The difference lies in the scale of measurement. As a rule, the ROC mid-point line is 50 (instead of a zero line), and the indicator fluctuates below and above 50 instead of 0, so there are no negative values. The Momentum and ROC are important signals measuring trend acceleration, i.e. whether the trend picks up speed or slows down. These tools act as leading indicators, reaching the highest level ahead of the trend and falling to the lowest mark before prices hit the bottom. As long as oscillators are moving towards new highs, a trader can stay long. When an oscillator reaches a higher peak, it signals that the bullish trend gains pace and is likely to stay on an upward path. After a lower high has been exceeded, the bullish trend loses steam, so a trader should prepare for a reversal. The same is true for lows in a bearish trend. The oscillators that we have discussed above involve the same drawback as simple MAs. They react to each day’s price data twice: first when they enter the window, and then when they leave it.

When these oscillators reach a new high, indicating increasing pessimism in the market, prices are most likely to continue an upward movement. If prices are rising while the values of oscillators are declining, this shows an upcoming price peak, so it is time to cash in profit from long positions and get ready for a reversal, and vice versa for a bearish trend.

When dealing with leading indicators, a trader should use caution and keep in mind the following recommendations:

  1. In an upward trend, buy whenever an oscillator starts rising after dropping below the zero line. This signal shows that the trend is slowing down. In a downward trend, sell if an oscillator rises above the zero line and then starts to decline.
  2. A new peak of an indicator reflects a high level of bullish enthusiasm, which is likely to push the market even higher. In this case, it is relatively safe to stay long. If declining peaks of an oscillator point to weaker bullish sentiment, reverse your position immediately. The opposite is true for a bearish trend.
  3. A break in the trend line of an oscillator most frequently precedes a break in the price trend line for one or two days. Therefore, prepare for a reversal whenever a leading indicator breaks a trend line.

An oscillator is said to be in an overbought condition when it reaches a higher level compared with its previous values. Overbought means that the market has gone up too far and the oscillator is ready to turn downwards. Overbought is a market condition when bulls are either unwilling or no longer able to buy, so they cannot push prices any higher. This is a relatively calm period in the market, with no clear bullish direction in prices. As bulls fail to push the price to a new level, and the price action n days ago (i.e. within the period where we take the price from, serving as a midpoint for oscillator calculation) has an upward slope, the oscillator reverses down and, after a while, breaks the overbought barrier. In such a way, most oscillators signal weaker bullish pressure and an upcoming trend reversal. An oversold condition occurs when an oscillator reaches a lower level compared with its previous values. In this case, the oscillator is ready for an upturn.
Overbought and oversold levels are marked with horizontal reference lines on a price chart. They are drawn in such a way that an oscillator spends no more than 5 percent of total time beyond each of these lines. These reference lines should only run across the highest peaks and the lowest dips of an oscillator for the past six months. These lines should be readjusted every three months.
An oscillator may stay in an overbought area for many weeks when a new bullish trend emerges, sending premature sell signals. Conversely, in a bearish trend an oscillator may remain in an overbought condition for weeks, giving false buy signals. In such cases, it may be a good idea for a chartist to switch to trend-following indicators.
As well as some of the technical tools we have discussed above, oscillators generate the most reliable signals when there is a certain discrepancy (i.e. divergence) between indicators and prices. A bullish (positive) divergence occurs when prices soar to a new high while the indicator traces a lower low. This situation means that prices are rising out of inertia, and bulls are losing control. The opposite is true for a bearish (negative) divergence. Keep in mind that triple divergences, though rare, may also occur in oscillators.

There are three main ways to use the Momentum oscillator:

  1. You can use the Momentum as a trend-following oscillator similar to the MACD. In this case, a buy signal occurs when the indicator forms a bottom and then starts rising. At the same time, a sell signal is in place when the indicator turns down after reaching a peak. To determine the oscillator’s reversal points more precisely, you can apply its short-term moving average.
  2. Extremely high or low values of the Momentum oscillator imply a continuation of the existing trend. For example, if the oscillator reaches extremely high values and then turns lower, you should expect prices to go even higher. In any case, do not rush to open or close a position until prices have confirmed the signal sent earlier by the oscillator.
  3. You can also use the Momentum as a leading indicator. This method is based on the assumption that the final phase of a bullish trend is usually accompanied by rapid price growth while the end of a bearish market is identified with a steep price drop. Even though this may often be the case indeed, it is still a broad generalization.

The Momentum and ROC function in a similar way, signaling a potential change of the trend direction after reaching the upper boundaries of the time frame.

RSI (Relative Strength Index)

The Relative Strength Index is a price-following oscillator that fluctuates in a range between 0 and 100. One of the most common techniques of RSI analysis is to search for divergences where the price makes a new high while the RSI fails to break above its previous peak. Such a divergence indicates a potential trend reversal. If the RSI then turns down and sinks below its bottom, it completes the so-called failure swing. This failure swing is considered to be a confirmation of a soon price reversal.

There are several applications of the RSI for chart analysis.

  1. Tops and bottoms
    The RSI usually forms peaks above 70 and bottoms below 30. As a rule, these tops and bottoms anticipate actual price swings on the underlying chart.
  2. Chart patterns
    The RSI often forms graphical patterns such as Head and Shoulders and Triangle which are not necessarily visible on a price chart.
    Failure swings (support or resistance breakouts) occur whenever the RSI rises above the previous high or falls below the prior low.
  3. Support and Resistance
    On an RSI chart, support and resistance levels are seen even more clearly than on a price chart.
  4. Divergences
    As we have mentioned above, divergences take place when the price makes a new high or low that is not confirmed by a new high or low on the RSI chart. Prices usually correct in the direction of the RSI.

The Relative Strength Index was invented and developed by Welles Wilder in the mid-1970s. The RSI is one of the most popular and widespread oscillators. When it comes to the RSI, apart from traditional oscillator analysis, chartists also apply graphical analysis with support, resistance, and trend lines as described above.
To calculate values of the oscillator, use the following formula:

RSI = 100 - [100 / (1 + RS)]; RS = AUx/ADx

where x is the number of days;

AU is an average of earlier prices that closed higher for x days;

AD is an average of earlier prices that closed lower for x days.

This leading or coincident indicator measures the strength of bullish or bearish sentiment in a predefined period by tracking changes in closing prices during that time. The RSI fluctuates between 0 and 100. Horizontal reference lines that divide overbought and oversold areas must run across the highest peaks and lowest bottoms. As a rule, they are drawn at the levels of 30 and 70. However, in strong trending markets these lines may be shifted to 40 and 80 (in an uptrend), or 20 and 60 (in a downtrend). The basic rule here may be formulated as: draw reference lines so that the RSI spends no longer than 5% of its time above the highest and below the lowest levels in the previous 4 to 6 months. These lines should be readjusted every three months. There are three types of trading signals generated by the RSI: divergences, RSI chart patterns, and levels.

Bullish (positive) divergences give buy signals. They take place whenever prices reach a new low while the RSI makes a higher bottom than the one recorded during its previous downturn. A trader should go long at the moment when the RSI starts going up from the recent bottom, and place a stop order below the most recent lowest price. A buy signal would be particularly strong if the preceding RSI bottom is below the lower reference line, and the most recent bottom is above this line.

The opposite is true in the case of bearish (negative) divergences. Likewise, a sell signal would be especially strong if the preceding RSI peak is above the upper reference line, and the most recent peak is below it. Out of all technical indicators, traditional graphical methods work best with the Relative Strength Index. The dynamic of the RSI chart precedes prices by several days, providing hints at further market behavior. The RSI trend line reverses one or two days before the price trend.

Rules for RSI trend line analysis:

  • — Place long orders when the RSI breaks below the falling RSI trend line.
  • — Place short orders when the RSI breaks above the rising RSI trend line.
  • — When the RSI rises above the upper reference line, it indicates a high level of bullish activity. At the same time, the market is in an overbought condition, so keep in mind that a wave of sell-offs is about to start.
  • — When the RSI falls below the lower reference line, we observe strong bearish pressure. Take into account that the market is then oversold, and traders will start making bullish bets in the short term.

ЗPlace long orders based on RSI overbought signals only when the weekly trend is rising.
There is one more important point to grasp when dealing with oscillators.
Remember that any strong trend, whether it is rising or falling, usually pushes an oscillator to extreme areas. In this case, indicators can give premature overbought or oversold signals, which may lead to an early exit from profitable positions. For example, in a strong uptrend, the market may stay in an overbought area for quite a long time. However, the mere fact that an oscillator hovers around extreme points does not necessarily mean that a trader needs to liquidate long positions immediately or open short orders in a strong uptrend. The first time when an oscillator has touched an overbought or oversold area is usually only a warning. However, if the curve returns to these critical levels once again, this is a more important signal that requires your full attention. If further moves fail to confirm an upcoming rise or fall, the oscillator forms a double top or bottom. This means that a divergence may possibly take place, so a trader is recommended to protect existing positions. If the curve turns in the opposite direction and crosses the previous peak or bottom, the divergence signal is confirmed. However, even an exit from a position may turn out to be premature, at least until there are any signs of a trend change.

The indicator’s exit from an oversold area supported by a bullish divergence has given a strong buy signal.

Stochastic Oscillator

The Stochastic oscillator was invented by George Lane in the 1950s. It is one of the most popular tools in technical analysis.

As we have already discussed, an oscillator in general represents a curve that fluctuates in a range from 0 to 100. If an oscillator rises above 70 or falls below 30, prices are said to be beyond the equilibrium. Buy when an oscillator falls below 30 and then rises above that level. Sell when it rises above 70 and then falls below that level. A simple moving average drawn on the stochastic oscillator chart may also be regarded as a signal line. Stochastic oscillators are the most frequently used indicators in technical analysis. This method helps to determine price movement on a predefined period.

%K = 100 * [Ct - L5 / H5 - L5]

where Ct is the most recent closing price;

L5 is the lowest level for the last 5 days;

H5 is the highest level for the last 5 days.

This formula allows a chartist to locate, on a percentage basis (from 0 to 100), the most recent closing price in relation to the total price range over a predefined time period. A value above 70 means that the closing price is near the upper end of the range. Accordingly, a low reading below 30 implies that the closing price is located around the lower end of the range.

The second %D curve is a three-day simple moving average of the %K line. The %D curve is calculated according to the following formula:

D% = 100 * CL3 / HL3

where CL3 is a three-day sum of (Ct - L5);

HL3 is a three-day sum of (H5 - L5).

These formulas are used for plotting two curves that fluctuate on a vertical scale from 0 to 100.The %K curve is displayed on charts as a solid line while the slower %D line is dashed.

The lines that we have discussed above are called fast Stochastics. Some traders prefer to use slow Stochastics. In this case, the formula would be slightly different. The %K line is calculated according to the %D formula while the %D line is calculated as a moving average of the %K line. Stochastic oscillators reflect the capacity of bulls or bears to close the market near the upper or lower end of the price range. In a rising market, the price tends to close near the upper boundary. If bulls can push prices higher within one day yet the price fails to close near the highest level, Stochastic lines begin moving lower, sending a sell signal. The opposite is also possible. The divergence between the %D line and the price that takes place when the curve reaches an overbought or oversold area presents another important signal for a chartist. These critical areas start beyond horizontal lines (above 70 and below 30). The most reliable buy and sell signals occur when the %D line is located within the 10 to 15 or 80 to 90 area accordingly.
The Stochastic oscillator gives three types of trading signals:

  1. The most powerful buy and sell signals occur when there is a divergence between prices and the indicator. A bearish divergence takes place when the %K curve is above 70, forming two falling peaks, while prices continue to grow. Conversely, in a bullish divergence, the %D curve is below 30, forming a double rising bottom, while prices extend their decline. When all of these factors are in place, a final buy or sell signal occurs when the % K line crosses the %D line after the latter has already reversed its direction. In other words, the crossing should take place on the right of the %D line critical levels. A buy signal is more significant in the lower area if the %K curve crosses over the %D curve to the upside after the %D curve has already turned upwards. A sell signal is more reliable in the upper area if the %D line has already reversed and started falling before it was crossed by the %K curve. Therefore, the crossing is more significant if both curves are moving in the same direction.
  2. Another signal worth paying attention to occurs when the Stochastic oscillator moves into an overbought or oversold area. In a bullish weekly trend, when daily Stochastic lines fall below their lower line, a trader should place a buy order with a protective stop near the previous bottom. In the opposite case, a sell order is recommended. The shape of the Stochastic’s extreme points often allows a chartist to predict whether the upturn will be strong or weak. When we observe a shallow and narrow bottom, bearish activity is rather weak, so the rally is expected to be quite strong. If the bottom is narrow and deep, it means that the rally is likely to be weak on the back of strong bearish activity. The same is true for the peaks of Stochastic lines.
  3. Another important signal to track is based on the relative direction of the price curve and stochastic lines. The Stochastic confirms short-term trends when both of its lines are heading in the same direction. The uptrend is most likely to persist if both prices and Stochastic lines are on the rise. Alternatively, if prices slide while both Stochastic lines climb, the short-term downtrend is expected to continue.

Note the reliability of the sell signals sent by the Stochastic on the USD/RUR chart. The %K line running across the %D line and then exiting the overbought territory to the downside gives a clear sell signal (given that we trade only in the direction of the market trend).

Helpful information for MetaTrader users

Below we have listed a number of indicators used in technical analysis that are available in the MetaTrader trading platform.

Average Directional Movement

One of the most common technical tools used to determine a market trend (i.e. the tendency of the financial asset’s price to move in a particular direction). This indicator was described in full detail by Welles Wilder in his book New Concepts in Technical Trading Systems.

Bollinger Bands

Bollinger Bands (also called standard deviation channels) represent lines (bands) plotted at levels equal to a certain amount of standard deviation from a moving average curve on a price chart. The width of standard deviation depends on market volatility. Bands widen in more turbulent market conditions and narrow during more stable periods.

Commodity Channel Index

This technical indicator is used to measure deviations of an asset’s price from its statistical historical average. When the index is high, the price is considered to be overvalued (unusually high) compared to its average. A low reading means that the price is undervalued (unusually low) compared to its average.


This oscillator reflects the relationship between the current price and the previous price bar. If the current high is above the previous high, the difference is calculated and recorded. If the current high is below or equal to the previous high, then the difference is either negative or equal, so the indicator is set to zero. The calculation is repeated over a series of consecutive days, and the sum of these values makes up the numerator of the DeMarker equation. The denominator is the numerator value plus the difference between price lows of the current and previous bars. A reading below 30 indicates a potential reversal to the upside; any value above 70 points means that price may reverse to the downside.


This technical indicator is composed of two moving average lines. One of the lines is shifted upwards; and the other, downwards. When the market moves beyond the limits of a trading range as a result of active sell-offs or purchases, the trend is most likely to reverse.

Moving Average

This tool has gained widespread recognition among technical analysts. MAs may be either used on their own or form the basis for other, more complex, trading indicators. Moving averages are measured over a predefined time period. Longer time frames reduce the probability of false signals but at the same time weaken the sensitivity of an MA to price data.


The MACD is an oscillator reflecting the difference between two exponential moving averages (EMAs) of closing prices. It is commonly calculated by subtracting a 26-day EMA from a 12-day EMA. However, the function actually uses exponential constants of 0.075 and 0.15, which are closer to 25.6667 and 12.3333 periods. A 9-period EMA of the MACD line is often used as a signal line on the MACD histogram.


This technical indicator measures the rate of an asset’s price fluctuations for a given period of time rather than prices themselves.


In general, an oscillator is calculated as a difference between a shorter-term moving average and a longer-term MA.

Parabolic SAR

The Parabolic SAR is an indicator plotted on a price chart that is similar to a moving average. The SAR stands for Stop and Reverse as this tool is used to identify points of potential trend stops and reversals. Unlike MAs, the Parabolic SAR moves with higher acceleration (referred to as the Acceleration Factor) and may change its position relative to prices. If the price breaks through the Parabolic SAR lines, the indicator signals a potential stop or reversal of a trend. In this case, further SAR values will be placed on the opposite side of price action. When a reversal takes place, the highest or lowest price over the previous period will act as a reference point. Such a reversal indicates either the end of a trend (i.e. an asset starts correcting or trading sideways) or its reversal.

Rate of Change

One of the most effective and simple oscillators that reflects the percentage change in price from one period to the next. The Rate of Change is measured by comparing the current price with the price n periods ago. Such periods can range from one minute to one month.

Relative Strength Index

The RSI is a technical indicator measuring the strength and speed of price changes. This oscillator is based on closing prices as it compares the average upward and downward price movements over a specified time period.

Standard Deviation

This indicator shows the degree of variation between the actual price and its average value over a certain time period. The Standard Deviation is calculated as the average square root of price deviation from its moving average.

Stohastic Oscillator

This tool compares the current closing price with the price range for a predefined period of time. The Stochastic Oscillator consists of two separate lines.

William's Percent Range

This is a dynamic indicator that determines overbought and oversold conditions. It represents a single line plotted on an upside-down scale.

The most sophisticated chartists can benefit from the opportunity to export MetaTrader data to specialized trading systems such as Omega TradeStation and MetaStock for more detailed technical analysis.

Questions for revision
  1. How would you define technical indicators?
  2. What groups of technical indicators are there?
  3. Which signal is the strongest in technical analysis? What specific features does it have?
  4. Describe trading signals given by the Stochastic oscillator.
  5. Which MACD signals indicate that a trader should open long positions?
  6. What does the ADX chart show?

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