**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.**

**Moving Averages**

**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**

**oscillator.**

**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**

**(green)**

**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**

**underlying currency.**

**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**

**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**

**index (RSI).**

**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**

**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**

**and**

**%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**

**%D line.**

**2. The right crossing, when the %K line occurs after the peak of the %D**

**line.**

**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**

**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**

**predetermined period.**

**The formula for calculating the momentum (M) is:**

**M=CCP-OCP, where**

**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:**

**Л5/=100-[100/(1+RS)], where**

**RS - (average of X days up closes/average of X days down**

**closes);**

**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**

**underlying price.**

**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**

**Williams %R.**

**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.)**