Market Indicators & Signal Mathematics
Deconstruct the mathematical equations powering technical analysis. No vague chart patterns—just formal statistical derivations, smoothing equations, and documented failure modes.
Relative Strength Index
Momentum oscillator measuring the speed and change of price movements
The Relative Strength Index (RSI) is an oscillator bounded between 0 and 100 developed by J. Welles Wilder Jr. in 1978. It measures the ratio of recent upward price changes to recent downward price changes to identify overbought or oversold conditions.
RSI = 100 - (100 / (1 + RS)), where RS = Average Gain / Average Loss over N periodsMoving Average Convergence Divergence
Trend-following momentum indicator displaying the relationship between two exponential moving averages
Developed by Gerald Appel in the late 1970s, MACD tracks the difference between a fast exponential moving average (typically 12-day) and a slow exponential moving average (typically 26-day), alongside a 9-day signal line and histogram.
MACD = EMA(12) - EMA(26); Signal = EMA(9) of MACD; Histogram = MACD - SignalSimple Moving Average
Arithmetic average of closing prices over a defined number of historical time periods
A Simple Moving Average (SMA) calculates the unweighted mean of closing prices over N periods. It smooths out short-term price volatility to expose underlying secular trends and dynamic institutional support/resistance levels.
SMA(N) = (Price_1 + Price_2 + ... + Price_N) / NExponential Moving Average
Weighted moving average applying exponentially decreasing weights to older price data
The Exponential Moving Average (EMA) addresses the lag of the SMA by applying a weighting multiplier k = 2 / (N + 1) to recent prices. This makes it more sensitive to recent market developments while maintaining historical trend continuity.
EMA(t) = Price(t) * k + EMA(t-1) * (1 - k), where k = 2 / (N + 1)Bollinger Bands
Dynamic volatility bands plotted at standard deviation intervals around a moving average
Invented by John Bollinger in the 1980s, Bollinger Bands consist of a middle band (typically 20-period SMA) and upper and lower bands positioned two standard deviations (2σ) away, dynamically expanding during high volatility and contracting during low volatility.
Upper Band = SMA(20) + (2 * StDev); Lower Band = SMA(20) - (2 * StDev)