Correlation coefficient, sometimes referred to simply as
correlation, refers to the degree of similarity between two variables. In the markets, correlation is typically used to measure how close the relationship is between two price series (e.g., two distinct stocks or markets), between an individual stock (or trading fund) and an index, and so on. Correlation coefficients range between -1.00 and +1.00, with +1.00 representing perfect positive correlation (i.e., two variables moving precisely in tandem); -1.00 represents perfect negative correlation (i.e., two variables moving exactly opposite to one another). A correlation coefficient of zero means the two variables have no discernible relation. The site
http://davidmlane.com/hyperstat/index.html offers relatively easy-to-digest definitions of this and other statistical terms.
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Elliott Wave is a descriptive form of technical analysis based on the concept that price action unfolds in identifiable, structured waves that define both trend and countertrend moves.
Ralph Nelson Elliott (1871-1948) introduced his ideas through a series of letters to Charles J. Collins, who help Elliott publish The Wave Principle in 1938. Also with Collins’ aid, Elliott published a series of articles in Financial World magazine in 1939. Today, Elliott Wave theory is probably best known through the work of Robert R. Prechter Jr., who in 1978 coauthored with A.J. Frost Elliott Wave Principle: Key to Stock Market Profits (John Wiley & Sons, 10th edition, 2001).
Elliott Wave theory contains elements of a concept known as a fractal, which is an object or shape that has self-similarity on different scales. Fractals are found in a variety of phenomena. For example, if you look at a mountain from a distance, you may see a peak with relatively smooth sides. As you move closer, you begin to see how the sides of the mountain are actually made up of smaller sub-peaks and sides, which consist of even smaller peaks and sides, all sharing a similar basic structure or pattern.
Similarly, part of wave theory is the idea that any wave cycle is part of a larger wave cycle that adheres to the same rules, and is also composed of smaller wave cycles with the same structure. Many devotees of Elliott Wave consider price action to be a natural phenomena driven by human emotion, which makes the fractal aspect of wave patterns a valid way to understand and describe the price movement.
Exponential moving average (EMA): While the simple moving average (SMA) calculation gives every price point in the average equal emphasis — for example, a five-day SMA is the sum of the most recent five closing prices divided by five — weighted moving averages emphasize more recent price action. An exponential moving average is a type of weighted moving average that uses the following formula:
EMA = SC * price + (1 - SC) * EMA(yesterday)
where: SC is a “smoothing constant” between 0 and 1, and EMA(yesterday) is the previous day’s EMA value.
You can approximate a particular SMA length for an EMA by using this formula to calculate the equivalent smoothing constant:
SC = 2/(n + 1)
where: n = the number of days in a simple moving average of approximately equivalent length.
For example, a smoothing constant of 0.095 creates an exponential moving average equivalent to a 20-day SMA (2/(20 + 1) = 0.095). The larger n is, the smaller the constant, and the smaller the constant, the less impact the most recent price action will have on the EMA. In practice, most software programs allow you to simply choose how many days you want in your moving average and select either simple, weighted, or exponential calculations.
Fibonacci series: A number progression in which each successive number is the sum of the two immediately preceding it: 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. As the series progresses, the ratio of a number in the series divided by the immediately preceding number approaches 1.618, a number that is attributed significance by many traders because of its appearance in natural phenomena (the progression of a shell’s spiral, for example), as well as in art and architecture (including the dimensions of the Parthenon and the Great Pyramid). The inverse, 0.618 (0.62), has a similar significance.
Some traders use fairly complex variations of Fibonacci numbers to generate price forecasts, but a basic approach is to use ratios derived from the series to calculate likely price targets. For example, if a stock broke out of a trading range and rallied from 25 to 55, potential retracement levels could be calculated by multiplying the distance of the move (30 points) by Fibonacci ratios –– say, 0.382, 0.50, and 0.618 –– and then subtracting the results from the high of the price move. In this case, retracement levels of 43.60 [55 - (30 * 0.38)], 40 [55 - (30 * 0.50)], and 36.40 [55 - (30 * 0.62)] would result.
Similarly, after a trading range breakout and an up move of 10 points, a Fibonacci follower might project the size of the next leg up in terms of a Fibonacci ratio –– e.g., 1.382 times the first move, or 13.82 points in this case.
The most commonly used ratios are 0.382, 0.50, 0.618, 0.786, 1.00, 1.382, and 1.618. Depending on circumstances, other ratios, such as 0.236 and 2.618, are used.
While Fibonacci retracements are used to calculate the possible partial correction levels of a previous price move (i.e., a reversal of up to 100 percent of a previous price swing), Fibonacci extension levels are used to extrapolate moves in the same direction as a previous price swing — for example, projecting a target for a new upswing that represents a 161.8-percent gain from a certain price level based on the size of the previous upswing.
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Keltner channels: Price bands formed by placing lines above and below a moving average of the “typical price” (the average of a bar’s high, low, and closing prices) by a distance equal to the moving average of the daily ranges over an n-day period. The following definition uses a 10-day simple moving average (SMA) for both calculations.
1. Upper Keltner channel: Today’s 10-day moving average of the typical price + 10-day SMA of the daily ranges (high-low).
2. Lower Keltner channel: Today’s 10-day moving average of the typical price - 10-day SMA of the daily ranges (high-low).
Linear regression (“best-fit”) line: A way to calculate a straight line that best fits a set of data (such as closing prices over a certain period) — that is, a line that most accurately reflects the slope, or trend, of the data.
A regression line is calculated using the “least squares” method, which refers to finding the minimum squared (x*x, or x2) differences between price points and a straight line. For example, if two closing prices are 2 and 3 points away (the distance being calculated vertically) from a straight line, the squared differences between the points and the line are 4 and 9, respectively.
The squared differences are used (instead of just the differences) because some differences are negative (for points below the line) and others are positive (for points above the line). Squaring all the differences creates all-positive values and allows you to calculate a formula for the straight line.
The “best-fit” line is the line for which the sum of the squared differences between each price and the straight line are minimized.
The formula for a straight line (y) is:
y = a + b*t
where:
t = time
a = the initial value of the line when “t” is equal to zero (sometimes called the “intercept” value — i.e., the point at which the line intercepts the vertical y-axis) or the point at which a specific line begins
b = the slope of the line, which is the rate at which the line rises or falls (e.g., 0.75 points per day).
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Moving average convergence-divergence (MACD): Although it is often grouped with oscillators, the MACD is more of an intermediate-term trend indicator (although it can reflect overbought and oversold conditions). The default MACD line (which can also be plotted as a histogram) is created by subtracting a 26-period exponential moving average (EMA) of closing prices from a 12-period EMA of closing prices; a nine-period EMA is then applied to the MACD line to create a “signal line.”
MACD = EMA(C,12)-EMA(C,26)
Signal line = EMA(MACD,9)
Moving average crossover: A trend-following approach that signals a trend change when a shorter-term moving average crosses above or below a longer-term moving average.
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Open interest: The number of outstanding (unclosed) positions in a given futures or options market.
Optimization (and walk-forward optimization, walk-forward testing, in-sample, and out-of-sample: The
process of testing a range of system parameters on historical price
data to find the "best" values for use in trading. For example, a
trading system that enters long when price makes an n-bar high might be
optimized to find the value of n, from 10 to 100, that produces the most
favorable results. There is a great deal of debate over the benefits of
optimization, because the process is often simply used to find system
parameter values that produce the highest net profit on a certain set of
historical data, with the expectation these values will produce similar
results in the future. However, this is rarely the case. Highly
optimized parameters typically do nothing more than reflect the specific
circumstances of the data set from which they were derived; they will
typically underperform in the future because price action never unfolds
exactly as it has in the past.
Some traders believe optimization of any kind is potentially damaging.
However, there is an argument that optimization can be used to find
less-idealized but more reliable parameter settings by identifying
ranges of values with comparable, positive results. In the case of the
aforementioned n-bar breakout system, it might turn out that a 23-bar
breakout produced the best results over the initial test period, but the
results for 20-, 21-, 22-, 24-, 25-, and 26-bar breakouts were all
negative, and even n values up to 30 produced sub-par results. This
suggests the 23-bar results were an anomaly, and using this parameter
would likely result in poor performance in the future. By contrast, if
all n values from 35 to 47 produced positive, comparable results — even
if they were much less profitable than the 23-bar breakout — a
representative value from this stable range (i.e., the median value, 41)
is much more likely to produce success in actual trading.
Typically, parameter values are optimized on an initial data set (the
"test," "sample" or "in-sample" data), and then tested on a new,
different data set (the "out-of-sample" data) to simulate the process of
trading the system. This can be repeated several times — the process of
"walking the system forward" on new data is the reason this approach is
referred to as "walk-forward" optimization. If the optimized values do
not perform consistently on new, out-of-sample data, they are not suited
for trading. Some system designers choose to re-optimize system
parameters on relatively short data periods and use these parameters on
the next set of data, believing the price behavior from the more recent
past has a greater chance of persisting into the near future. Others
believe this simply results in a highly optimized system that will
always be one step behind the market.
Finally, among the many problems of optimization is the challenge of
determining what constitutes a "best-performing" value. Traders are apt
to look exclusively at net profit, without incorporating measures of
risk (e.g., drawdown depth and duration) and consistency (e.g., winning
percentage, average trade) that are likely to make future profitability
more likely.
Purchasing power parity: The idea that an exchange rate should reflect the level that results in the same price (in the two currencies) for a product purchased in two countries. For example, if a certain automobile costs 50,000 British pounds in Great Britain, it should cost 25,000 U.S. dollars in the United States if the current British pound/U.S. dollar rate (GBP/USD) is 2.0000.
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Quantitative easing is a tool a central bank uses to attempt to stimulate the economy when cutting interest rates is not feasible — such as when rates are already at or near zero. Through quantitative easing, the central bank purchases assets (e.g., treasuries, mortgages, securities) from financial institutions to pump money into the financial system. Quantitative easing is often referred to as “printing money.” Critics contend the practice runs a high risk of creating high inflation, among other drawbacks.
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Simple moving average (SMA). The simple moving average (SMA) is the mean price of a stock, futures contract, currency, or other instrument over a specific time period:
N-day moving average = Sum (Pricet, Pricet-1 … Pricet-N)/N
where,
Pricet = today’s price
Pricet-N = price N days ago
For example, a 20-day SMA is the sum of the prices of the most recent 20 days, divided by 20.
The closing price is usually used in moving average calculations, although the high, low, opening price, or average price of a price bar is sometimes substituted. As the market moves forward in time, the newest price is added to the average and the oldest is dropped from it.
Moving averages can be calculated for any time increment – daily, intraday, weekly, monthly. For a five-minute bar chart, for example, a 10-bar SMA would be the average price of the 10 most recent five-minute bars. Moving averages are typically used to highlight trend direction and filter out smaller fluctuations (or “noise”) from a data series. The degree to which they do this depends on the length of the average: A longer moving average reflects a longer-term trend and filters out shorter-term fluctuations, while a shorter moving average reflects the more immediate trend and filters out less noise. However, this filtering comes at a cost: The longer the average, the more it lags behind the movements of the data series.
The European currency “snake:” A short-lived agreement among European member states in the early 1970s to limit currency fluctuations within a proscribed percentage band.
True range (TR): A measure of price movement or volatility that accounts for the gaps that occur between price bars. This calculation provides a more accurate reflection of the size of a price move over a given period than the standard range calculation, which is simply the high of a price bar minus the low of a price bar. The true range calculation was developed by Welles Wilder and discussed in his book New Concepts in Technical Trading Systems (Trend Research, 1978).True range can be calculated on any time frame or price bar — five-minute, hourly, daily, weekly, etc. Using daily price bars as an example, true range is the greatest (absolute) distance of the following:
1. Today’s high and today’s low.
2. Today’s high and yesterday’s close.
3. Today’s low and yesterday’s close.
Average true range (ATR) is simply a moving average of the true range over a certain time period. For example, the five-day ATR would be the average of the true range calculations over the last five days.
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Ulcer Index: A measure of the depth and duration of a drawdown from an equity peak.
Variance and standard deviation: Variance measures how spread out a group of values are — in other words, how much they vary. Mathematically, variance is the average squared “deviation” (or difference) of each number in the group from the group’s mean value, divided by the number of elements in the group. For example, for the numbers 8, 9, and 10, the mean is 9 and the variance is:
{(8-9)2 + (9-9)2 + (10-9)2}/3 = (1 + 0 + 1)/3 = 0.667
Now look at the variance of a more widely distributed set of numbers: 2, 9, and 16:
{(2-9)2 + (9-9)2 + (16-9)2}/3 = (49 + 0 + 49)/3 = 32.67
The more varied the prices, the higher their variance — the more widely distributed they will be. The more varied a market’s price changes from day to day (or week to week, etc.), the more volatile that market is.
A common application of variance in trading is standard deviation, which is the square root of variance. The standard deviation of 8, 9, and 10 is: sq. rt. 0.667 = .82; the standard deviation of 2, 9, and 16 is: sq. rt. 32.67 = 5.72.
Volatility: The level of price movement in a market. Historical (“statistical”) volatility measures the price fluctuations (usually calculated as the standard deviation of closing prices) over a certain time period — e.g., the past 20 days. Implied volatility is the current market estimate of future volatility as reflected in the level of option premiums. The higher the implied volatility, the higher the option premium.
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