📉 Technical Indicators
This page documents the technical analysis indicators available as chart overlays in LibreFolio's FX module. Each indicator is explained from two complementary perspectives: the financial interpretation that traders use daily, and the signal processing equivalent that engineers from control systems or DSP backgrounds will recognise instantly.
Why two perspectives?
Financial markets are not stationary LTI (Linear Time-Invariant) systems — they are noisy, chaotic, and their spectral content shifts over time. Yet the mathematical tools we apply to extract trend, momentum, or volatility are exactly the same discrete-time filters taught in any signal processing course. If you have ever designed a Butterworth low-pass or computed a running variance, you already understand these indicators — just under different names.
⚡ The "Fast" vs "Slow" Intuition
In finance, fast and slow refer to the time constant (\(\tau\)) of the underlying filter.
| Property | Fast (small \(N\)) | Slow (large \(N\)) |
|---|---|---|
| Cut-off frequency \(f_c\) | Higher | Lower |
| Noise rejection | Poor — lets HF through | Good — strong smoothing |
| Phase lag | Small — reacts quickly | Large — significant delay |
| Typical \(N\) | 9, 12, 14 | 26, 50, 200 |
📉 EMA — Exponential Moving Average
💡 Financial Meaning
The EMA tracks the trend by smoothing daily price noise, giving more weight to recent observations than older ones. Traders overlay EMAs of different periods on a price chart: when a short-period EMA crosses above a long-period EMA, it signals upward momentum (a "golden cross"); the opposite crossing signals a slowdown ("death cross").
🔢 Mathematical Formula
The EMA is defined by the first-order recurrence:
where \(P_t\) is the closing price at time \(t\) and \(\alpha\) is the smoothing coefficient.
Mapping \(N\) → \(\alpha\). Traders specify a "period" \(N\) (in days). The coefficient is derived by matching the average age of data between an EMA and a Simple Moving Average (SMA) of the same window:
Setting them equal:
For example, \(N = 14 \implies \alpha = 2/15 \approx 0.133\).
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Period (\(N\)) | period |
14 | Lookback window in days. Higher → smoother, slower. |
| Offset | offset |
0 | Vertical shift as % of base value. |
🎛️ Signal Processing Equivalent — First-Order IIR Low-Pass Filter
The recurrence \(y[n] = \alpha\,x[n] + (1-\alpha)\,y[n-1]\) is precisely a first-order IIR (Infinite Impulse Response) low-pass filter. Its transfer function in the \(z\)-domain is:
The \(-3\,\text{dB}\) cut-off frequency (normalised) is:
When \(\alpha\) is small (\(N\) large) the pass-band narrows dramatically, attenuating all but the DC component (the long-run trend).
Pole location
The single pole sits at \(z = 1-\alpha\). For \(N = 200\), \(\alpha \approx 0.01\), so the pole is at \(z = 0.99\) — extremely close to the unit circle, which explains the heavy smoothing and large group delay.
📊 MACD — Moving Average Convergence Divergence
💡 Financial Meaning
The MACD answers: "Is the trend accelerating or losing steam?" It does not tell you the price is rising (you can see that already); it tells you whether the rate of change of the trend is positive or negative. Traders watch for the MACD line crossing the Signal line — a bullish crossover suggests increasing momentum, a bearish one suggests exhaustion.
🔢 Mathematical Formulas
The MACD system produces three series:
-
MACD Line (the band-pass output):
\[ MACD_t = EMA_{fast}(C_t) - EMA_{slow}(C_t) \] -
Signal Line (smoothed MACD):
\[ Signal_t = EMA_{signal}(MACD_t) \] -
Histogram (momentum delta):
\[ Histogram_t = MACD_t - Signal_t \]
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Fast Period | fastPeriod |
12 | Short-term EMA window (days). |
| Slow Period | slowPeriod |
26 | Long-term EMA window (days). |
| Signal Period | signalPeriod |
9 | EMA smoothing applied to the MACD line. |
🎛️ Signal Processing Equivalent — Band-Pass Filter (Smoothed Derivative)
Subtracting two low-pass filters with different cut-off frequencies produces a band-pass filter. \(EMA_{fast} - EMA_{slow}\) cancels the DC component (the long-run trend shared by both) and suppresses high-frequency noise (already filtered by both EMAs). What remains is the mid-frequency band: the momentum oscillation.
In the \(z\)-domain:
The Signal Line is yet another low-pass applied to this band-pass output — it acts as a matched filter, delaying the signal slightly to reduce false-positive crossover detections.
Derivative interpretation
For small \(\alpha\), \(EMA_{fast} - EMA_{slow}\) behaves like a smoothed first derivative \(\frac{d}{dt}[\text{trend}]\). When the histogram flips sign, the "velocity" of the trend changes direction.
💪 RSI — Relative Strength Index
💡 Financial Meaning
The RSI measures whether buyers or sellers have dominated recently. It answers: "Over the last \(N\) days, how much of the total price movement was upward vs downward?" The result is squeezed into a 0–100 range:
- RSI > 70 → Overbought — the spring is stretched, a pullback is statistically likely.
- RSI < 30 → Oversold — the spring is compressed, a bounce is likely.
🔢 Mathematical Formulas
-
Decompose daily changes into gains and losses:
\[ U_t = \max(P_t - P_{t-1},\; 0), \qquad D_t = \max(P_{t-1} - P_t,\; 0) \] -
Smooth each component with an exponential moving average (SMMA variant):
\[ \overline{U} = SMMA_N(U), \qquad \overline{D} = SMMA_N(D) \] -
Relative Strength ratio and normalisation:
\[ RS = \frac{\overline{U}}{\overline{D}}, \qquad RSI = 100 - \frac{100}{1 + RS} \]
The normalisation \(100 - 100/(1+RS)\) is a monotonically increasing sigmoid that maps \(RS \in [0, \infty)\) to \(RSI \in [0, 100)\).
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Period (\(N\)) | period |
14 | Lookback window for SMMA. |
| Overbought | overbought |
70 | Threshold for overbought zone. |
| Oversold | oversold |
30 | Threshold for oversold zone. |
🎛️ Signal Processing Equivalent — Duty Cycle / Saturation Indicator
Imagine splitting the price delta signal \(\Delta P[n]\) into its positive and negative half-wave rectified components, then low-pass filtering each. The RSI is the ratio of the positive envelope to the total envelope, rescaled to \([0, 100]\).
In control systems terms, it is a saturation detector: when the system output (price) has been moving in one direction for too long, the RSI signals that the actuator (market) is near its rail. Like any oscillator in a feedback loop, the further from equilibrium, the stronger the restoring force — hence the mean-reverting property traders exploit.
Non-stationarity
The 70/30 thresholds assume roughly symmetric return distributions. In strong trending markets the RSI can stay above 70 for weeks — it is a probabilistic indicator, not a deterministic one.
📏 Bollinger Bands
💡 Financial Meaning
Bollinger Bands dynamically measure volatility and draw an adaptive "normality fence" around the price. When the bands are wide, the market is volatile; when they squeeze together, a breakout is imminent. A price touching the upper band signals statistical exuberance; touching the lower band signals an abnormal dip.
🔢 Mathematical Formulas
-
Middle Band (expected value):
\[ MB_t = SMA_N(C_t) \] -
Standard deviation of prices over the window:
\[ \sigma_t = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (C_{t-i} - MB_t)^2} \] -
Upper and Lower Bands:
\[ Upper_t = MB_t + k \cdot \sigma_t, \qquad Lower_t = MB_t - k \cdot \sigma_t \]
With \(k = 2\), if returns were normally distributed the price would stay inside the bands ~95.4% of the time. In practice, financial returns have fat tails (leptokurtosis), so breaches are more frequent — but still statistically significant.
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Period (\(N\)) | period |
20 | SMA window for expected value. |
| Multiplier (\(k\)) | multiplier |
2 | Number of standard deviations. |
🎛️ Signal Processing Equivalent — Adaptive Confidence Interval Tracker
The Middle Band is a FIR (Finite Impulse Response) moving average filter — the simplest low-pass with a rectangular window of length \(N\). The bands add a time-varying envelope at \(\pm k\sigma\), which is essentially a running estimate of the signal's instantaneous variance.
In the language of adaptive filters, this is an expected-value tracker with an adaptive confidence interval. When the variance \(\sigma^2\) drops (the "Bollinger Squeeze"), the system is in a low-entropy state. In chaotic systems like financial markets, low-entropy periods are reliably followed by high-entropy (high-volatility) explosions — making the squeeze one of the most watched setups in technical analysis.
FIR vs IIR
Unlike the EMA (IIR, one pole), the SMA is a FIR filter with a perfectly flat group delay of \((N-1)/2\) samples. It trades off a wider transition band for zero-phase distortion — ideal for centring the confidence envelope.