The mathematical trading methods provide a more objective view of
price activity. In addition, these methods tend to provide signals prior to their
occurrence on the currency charts. The tools of the mathematical methods
are moving averages and oscillators.
A moving average is an average of a predetermined number of prices
over a number of days, divided by the number of entries. The higher the
number of days in the average, the smoother the line is. A moving average
makes it easier to visualize currency activity without daily statistical noise. It
is a common tool in technical analysis and is used either by itself or as an
As one can see from Figure 5.35., a moving average has a smoother
line than the underlying currency. The daily closing price is commonly
included in the moving averages. The average may also be based on the
midrange level or on a daily average of the high, low, and closing prices.
Figure 5.35. Examples of three simple moving averages—5-day (white), 20-day (red) and 60-day
It is important to observe that the moving average is a follower rather
than a leader. Its signals occur after the new movement has started, not before.
There are three types of moving averages:
1. The simple moving average or arithmetic mean.
2. The linearly weighted moving average.
3. The exponentially smoothed moving average.
As described, the simple moving average or arithmetic mean is the
average of a predetermined number of prices over a number of days, divided
by the number of entries.
Traders have the option of using a linearly weighted moving average
(See Figure 5.36.). This type of average assigns more weight to the more
recent closings. This is achieved by multiplying the last day's price by one,
and each closer day by an increasing consecutive number. In our previous
example, the fourth day's price is multiplied by 1, the third by 2, the second
by 3, and the last one by 4; then the fourth day's price is deducted. The new
sum is divided by 9, which is the sum of its multipliers.
Figure 5.36. Example of a 20-day simple moving average (red) as compared to a 20-day
weighted moving average (white)
The most sophisticated moving average available is the exponentially
smoothed moving average. (See Figure 5.37.) In addition to assigning
different weights to the previous prices, the exponentially smoothed moving
average also takes into account the previous price information of the
Figure 5.37. Example of a 20-day simple moving average (red) as compared to a 20-day
exponential moving average (white)
Trading Signals of Moving Averages
Single moving averages are frequently used as price and time filters. As
a price filter, a short-term moving average has to be cleared by the currency
closing price, the entire daily range, or a certain percentage (chosen at the
discretion of the trader).
The envelope model (See Figure 5.38.) serves as a price filter. It
consists of a short-term (perhaps 5-day) closing price based moving average
to which a small percentage (2 percent is suggested for foreign currencies.)
are added and substracted. The two winding parallel lines above and below
the moving average will create a band bordering most price fluctuations.
When the upper band is penetrated, a selling signal occurs. When the lower
band is penetrated, a buying signal occurs. Because the signals generated by
the envelope model are very short-term and they occur many times against
the ongoing direction of the market, speed of execution is paramount. The
high-low band is set up the same way, except that the moving average is
based on the high and low prices. As a time filter, a short number of days
may be used to avoid any false signals.
Figure 5.38. An envelope model define the edges of the band. A close above the upper
band sends a buying signal and one below the lower band gives a selling signal
Usually traders choose a number of averages to use with a currency. A
suggested number is three, as more signals may be available. It may be
helpful to use intervals that better encompass short-term, medium-term, and
long-term periods, to arrive at a more complex set of signals. Some of the
more popular periods are 4, 9, and 18 days; 5, 20, and 60 days; and 7, 21,
and 90 days. Unless you focus on a specific combination of moving averages
(for instance, 4, 9, and 18 days), the exact number of days for each of the
averages is less important, as long as they are spaced far enough apart from
each other to avoid insignificant signals.
A buying signal on a two-moving average combination occurs when the
shorter term of two consecutive averages intersects the longer one upward. A
selling signal occurs when the reverse happens, and the longer of two
consecutive averages intersects the shorter one downward. (See Figure 5.39.)
Oscillators are designed to provide signals regarding overbought and
oversold conditions. Their signals are mostly useful at the extremes of their scales
and are triggered when a divergence occurs between the price of the underlying
currency and the oscillator. Crossing the zero line, when applicable, usually
generates direction signals. Examples of the major types of oscillators are moving
averages convergence-divergence (MACD), momentum and relative strength
Figure 5.39. Examples of a sell signal (first and third crossovers) and a buy signals (second
crossover) provided by the 5-day (red) and 20-day (white) moving averages
Stochastics generate trading signals before they appear in the price
itself. Its concept is based on observations that, as the market gets high, the
closing prices tend to approach the daily highs; whereas in a bottoming market,
the closing prices tend to draw near the daily lows.
The oscillator consists of two lines called %K and %D. Visualize %K as the
plotted instrument, and %D as its moving average.
The formulas for calculating the stochastics are:
%K = [(CCL -L9)I(H9 - L9)] * 100, where
CCL = current closing price
L9 - the lowest low of the past 9 days
H9 - the highest high of the past 9 days
%D=(H3/L3~) * 100,
where H3 = the three-day sum of (CCL - L9)
L3 = the three-day sum of (H9 - L9)
The resulting lines are plotted on a 1 to 100 scale, with overbought and
oversold warning signals at 70 percent and 30 percent, respectively. The buying
(bullish reversal) signals occur under 10 percent, and conversely the selling
(bearish reversal) signals come into play above 90 percent after the currency
turns. (See Figure 5.40.) In addition to these signals, the oscillator-currency price
divergence generates significant signals.
Figure 5.40. An example of the stochastic
The intersection of the %D and %K lines generates further trading signals.
There are two types of intersections between the %D and %K lines:
1. The left crossing, when the %K line crosses prior to the peak of the
2. The right crossing, when the %K line occurs after the peak of the %D
Moving Average Convergence-Divergence (MACD)
The moving average convergence-divergence (MACD) oscillator,
developed by Gerald Appel, is built on exponentially smoothed moving aver
ages. The MACD consists of two exponential moving averages that are plotted
against the zero line. The zero line represents the times the values of the two
moving averages are identical.
In addition to the signals generated by the averages' intersection with
the zero line and by divergence, additional signals occur as the shorter
average line intersects the longer average line. The buying signal is displayed
by an upward crossover, and the selling signal by a downward crossover.
(See Figure 5.41.)
Figure 5.41. An example of MACD
Momentum is an oscillator designed to measure the rate of price
change, not the actual price level. This oscillator consists of the net difference
between the current closing price and the oldest closing price from a
The formula for calculating the momentum (M) is:
CCP - current closing price
OCP - old closing price for the predetermined period.
The new values thus obtained will be either positive or negative
numbers, and they will be plotted around the zero line. At extreme positive
values, momentum suggests an overbought condition, whereas at extreme
negative values, the indication is an oversold condition. (See Figure 5.42.)
The momentum is measured on an open scale around the zero line.
Figure 5.42. An example of the momentum oscillator
This may create potential problems when a trader must figure out
exactly what an extreme overbought or oversold condition means. On the
simplest level, the relativity of the situation may be addressed by analyzing
the previous historical data and determining the approximate levels that
delineate the extremes. The shorter the number of days included in the
calculations, the more responsive the momentum will be to short-term
fluctuations, and vice versa. The signals triggered by the crossing of the zero
line remain in effect. However, they should be followed only when they are
consistent with the ongoing trend.
The Relative Strength Index (RSI)
The relative strength index is a popular oscillator devised by Welles
Wilder. The RSI measures the relative changes between the higher and lower
closing prices. (See Figure 5.43.)
Figure 5.43. An example of the RSI oscillator
The formula for calculating the RSI is:
RS - (average of X days up closes/average of X days down
X - predetermined number of days The original number of
days, as used by its author, was 14 days. Currently, a 9-day
period is more popular.
The RSI is plotted on a 0 to 100 scale. The 70 and 30 values are used
as warning signals, whereas values above 85 indicate an overbought
condition (selling signal) and values under 15 indicate an oversold condition
(buying signal.) Wilder identified the RSI's forte as its divergence versus the
Rate of Change (ROC)
The rate of change is another version of the momentum oscillator. The
difference consists in the fact that, while the momentum's formula is based
on subtracting the oldest closing price from the most recent, the ROC's
formula is based on dividing the oldest closing price into the most recent one.
(See Figure 5.44.)
Figure 5.44. An example of the rate of change (ROC) oscillator
ROC = (CCP/OCP) * 100, where
CCP - current closing price;
OCP = old closing price for the predetermined period Larry
The Larry Williams %R
The Larry Williams %R is a version of the stochastics oscillator. It
consists of the difference between the high price of a predetermined number
of days and the current closing price, which difference in turn is divided by
the total range. This oscillator is plotted on a reversed 0 to 100 scale.
Therefore, the bullish reversal signals occur at under 80 percent, and the
bearish signals appear at above 20 percent. The interpretations are similar to
those discussed under stochastics. (See Figure 5.45.)
Commodity Channel Index (CCI)
The commodity channel index was developed by Donald Lambert. It
consists of the difference between the mean price of the currency and the
average of the mean price over a predetermined period of time (See Figure
5.46.). A buying signal is generated when the price exceeds the upper (+100)
line, and a selling signal occurs when the price dips under the lower (-100)
line. (See Figure 5.46.)