Mathematical Trading Indicators

The mathematical trading methods provide an objective view of price activity. It helps you to build up a view on price direction and timing, reduce fear and avoid overtrading. Furthermore, these methods tend to provide signals of price movements prior to their occurring in the market.
The tools used by the mathematical trading methods are moving averages and oscillators. (Oscillators are trading tools that offer indications of when a currency is overbought or oversold). Though there are countless mathematical indicators, here we will cover only the most important ones.

1. Simple and Exponential Moving Average (SMA - EMA)
2. Moving Average Convergence-Divergence (MACD)
3. Bollinger Bands
4. The Parabolic System, Stop-and-Reverse (SAR)
5. RSI (Relative Strength Index)

Moving Average

A moving average is an average of a shifting body of prices calculated over a given number of days. A moving average makes it easier to visualize market trends as it removes – or at least minimizes - daily statistical noise. It is a common tool in technical analysis and is used either by itself or as an oscillator.
There are several types of moving averages, but we will deal with only two of them: the simple moving average (SMA) and the exponential moving average (EMA).

1. Simple moving average (SMA)
   • Definition The simple moving average is an arithmetic mean of price data. It is calculated by summing up each interval's price and dividing the sum by the number of intervals covered by the moving average. For instance, adding the closing prices of an instrument for the most recent 25 days and then dividing it by 25 will get you the 25 day moving average.
     Though the daily closing price is the most common price used to calculate simple moving averages, the average may also be based on the midrange level or on a daily average of the high, low, and closing prices.
     
   • Advantages Moving average is a smoothing tool that shows the basic trend of the market.
     It is one of the best ways to gauge the strength a long-term trend and the likelihood that it will reverse. When a moving average is heading upward and the price is above it, the security is in an uptrend. Conversely, a downward sloping moving average with the price below can be used to signal a downtrend.
     
   • Drawbacks It is a follower rather than a leader. Its signals occur after the new movement, event, or trend has started, not before. Therefore it could lead you to enter trade some late.
     It is criticized for giving equal weight to each interval. Some analysts believe that a heavier weight should be given to the more recent price action.
     
   • Example You can see from the chart below examples of two simple moving averages - 5 days (Red), 20 days (blue).
     
     
2. Exponential Moving Average (EMA) The exponential moving average (EMA) is a weighted average of a price data which put a higher weight on recent data point.
   
   • Characteristics The weighting applied to the most recent price depends on the specified period of the moving average. The shorter the EMA period, the more weight will be applied to the most recent price.
     An EMA can be specified in two ways: as a percentage-based EMA, where the analyst determines the percentage weight of the latest period's price, or a period-based EMA, where the analyst specifies the duration of the EMA, and the weight of each period is calculated by formula. The latter is the more commonly used.
     
   • Main Advantages compared to SMA Because it gives the most weight to the most recent observations, EMA enables technical traders to react faster to recent price change.
     As opposed to Simple Moving Average, every previous price in the data set is used in the calculation of EMA. While the impact of older data points diminishes over time, it never fully disappears. This is true regardless of the EMA's specified period. The effects of older data diminish rapidly for shorter EMAs than for longer ones but, again, they never completely disappear.
     
   • Example You can see from the chart below the difference between SMA (in blue) and EMA (in green) calculated over a 20-day period.
     
     

MACD (Moving Average Convergence-Divergence)

The moving average convergence-divergence indicator (MACD) is used to determine trends in momentum.

• Calculation It is calculated by subtracting a longer exponential moving average (EMA) from a shorter exponential moving average. The most common values used to calculate MACD are 12-day and 26-day exponential moving average.
  Based on this differential, a moving average of 9 periods is calculated, which is named the "signal line".
  MACD = [12-day moving average – 26-day moving average] > Exponential Weighted Indicator
  Signal Line = Moving Average (MACD) > Average Weighted Indicator
  
• Interpretation Due to exponential smoothing, the MACD Indicator will be quicker to track recent price changes than the signal line. Therefore,
  When the MACD crossed the SIGNAL LINE: the faster moving average (12-day) is higher than the rate of change for the slower moving average (26-day). It is typically a bullish signal, suggesting the price is likely to experience upward momentum.
  Conversely, when the MACD is below the SIGNAL LINE: it is a bearish signal, possibly forecasting a pending reversal.
  
• Example of a MACD You can see from the chart below example of a MACD. The MACD Indicator is represented in green and the Signal Line in Blue.
  
  

Bollinger Bands

Bollinger Bands were developed by John Bollinger in the early 1980s. They are used to identify extreme highs or lows in price. Bollinger recognized a need for dynamic adaptive trading bands, whose spacing varies based on the volatility of the prices. During period of high volatility, Bollinger bands widen to become more forgiving. During periods of low volatility, they narrow to contain prices.

• Calculation Bollinger Bands consist of a set of three curves drawn in relation to prices:
  The middle band reflects an intermediate-term trend. The 20 day - simple moving average (SMA) usually serves this purpose.
  The upper band is the same as the middle band, but it is shifted up by two standard deviations, a formula that measures volatility, showing how the price can vary from its true value
  The lower band is the same as the middle band, but it is shifted down by two standard deviations to adjust for market volatility.
  Bollinger Bands establish a Bandwidth, a relative measure of the width of the bands, and a measure of where the last price is in relation to the bands.
  
  Lower Bollinger Band = SMA - 2 standard deviations
  Upper Bollinger Band = SMA + 2 standard deviations.
  Middle Bollinger Band = 20 day - simple moving average (SMA).
  
• Interpretation The probability of a sharp breakout in prices increases when the bandwidth narrows.
  When prices continually touch the upper Bollinger band, the prices are thought to be overbought; triggering a sell signal.
  Conversely, when they continually touch the lower band, prices are thought to be oversold, triggering a buy signal.
  
• Example of Bollinger Bands You can see from the chart below the Bollinger Bands of the S&P 500 Index, represented in green.
  
  

The Parabolic System, Stop-and-Reverse (SAR)

The parabolic SAR system is an effective investor's tool that was originally devised by J. Welles Wilder to compensate for the failings of other trend-following systems.

• Description The Parabolic SAR is a trading system that calculates trailing "stop-losses" in a trending market. The chart of these points follows the price movements in the form of a dotted line, which tends to follow a parabolic path.
  
• Interpretation When the parabola follows along below the price, it is providing buy signals.
  When the parabola appears above the price, it suggests selling or going short.
  The “stop-losses” dots are setting the levels for the trailing stop-loss that is recommended for the position. In a bullish trend, a long position should be established with a trailing stop that will move up every day until activated by the price falling to the stop level. In a bearish trend, a short position can be established with a trailing stop that will move down every day until activated by the price rising to the stop level.
  The parabolic system is considered to work best during trending periods. It helps traders catch new trends relatively early. If the new trend fails, the parabola quickly switches from one side of the price to the other, thus generating the stop and reverse signal, indicating when the trader should close his position or open an opposing position when this switch occurs.
  
• Example of an SAR parabolic study You can see from the chart below in green the Parabolic System applied to the USDJPY pair.
  
  

Relative Strength Index (RSI)

The RSI was developed by J. Welles Wilder as a system for giving actual buy and sell signals in a changing market.

• Definition RSI is based on the difference between the average of the closing price on up days vs. the average closing price on the down days, observed over a 14-day period. That information is then converted into a value ranging from 0 to 100.
  When the average gain is greater than the average loss, the RSI rises, and when the average loss is greater than the average gain, the RSI declines.
  
• Interpretation The RSI is usually used to confirm an existing trend. An uptrend is confirmed when RSI is above 50 and a downtrend when it's below 50.
  It also indicates situations where the market is overbought or oversold by monitoring the specific levels (usually “30” and “70”) that warn of coming reversals.
  An overbought condition (RSI above 70) means that there are almost no buyers left in the market, and therefore prices are more likely to decline as those who previously bought will now take their profit by selling.
  An oversold condition (RSI below 30) is the exact opposite.
  
• Example of RSI You can see in red from the chart below the Relative Strength Index of the GBPUSD pair.