📉 EMA — Exponential Moving Average

The EMA tracks the trend by smoothing daily price noise, giving more weight to recent observations than older ones.

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💡 Financial Meaning

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").

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🔢 Mathematical Formula

The EMA is defined by the first-order recurrence:

\[ EMA_t = \alpha \cdot P_t + (1 - \alpha) \cdot EMA_{t-1} \]

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:

\[ \text{Age}_{SMA} = \frac{N-1}{2}, \qquad \text{Age}_{EMA} = \frac{1-\alpha}{\alpha} \]

Setting them equal:

\[ \alpha = \frac{2}{N+1} \]

For example, \(N = 14 \implies \alpha = 2/15 \approx 0.133\).

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⚙️ 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.

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🎛️ 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:

\[ H(z) = \frac{\alpha}{1 - (1-\alpha)\,z^{-1}} \]

The \(-3\,\text{dB}\) cut-off frequency (normalised) is:

\[ \omega_c = \cos^{-1}\!\left(1 - \frac{\alpha^2}{2(1-\alpha)}\right) \]

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.

EMA on Wikipedia