Trading Indicator

I will write about trading indicators in this post.

Both using technical analysis and fundamental analysis can help traders make informed decisions.

I will cover various types of indicators, with elaborate and fundamental math definitions.

Data types

Granularity	Typical Schema	Common Industry Term	Key Contents	Typical Uses
MBO (Market‑By‑Order, L3)	Schema::Mbo	Full order‑book depth / Level 3	Add, cancel, and execution events for every individual order, including order_id	Market making, trade‑match replay, liquidity research
MBP‑10 (Market‑By‑Price, L2)	Schema::Mbp10	10‑level price depth	Aggregated order size and updates at each of the top 10 price levels	Order‑book dynamics, arbitrage modeling
MBP‑1 / TBBO (Top Book, L1)	Schema::Mbp1	Best bid/ask	Best bid and ask quotes plus accompanying trades	Basic quote display, NBBO comparison
Trades	Schema::Trades	Tick‑by‑tick trades / last sale	Every executed trade (price, size, aggressor side, etc.)	Price‑volume analysis, VWAP, trade‑driven strategies
OHLCV‑T	Schema::Ohlcv	Bars / K‑line	Aggregated O‑H‑L‑C‑V data per second, minute, hour, or day	Backtesting, charting, low‑frequency signals

We now only use OHLCV‑T data, the other data types are used for high-frequency trading and are not available in our current setup.

Moving Averages (MA)

Simple Moving Average (SMA)

For a price (or data) series $P_t$ and window length $N$:

$$\text{SMA}_N(t)=\frac{1}{N}\sum_{i=0}^{N-1} P_{t-i}$$

• $t$ — index of the current bar (latest observation)
• $P_{t-i}$ — value $i$ bars ago
• $N$ — number of observations in the averaging window

indicator("ta.sma")

pine_sma(x, N) =>
    sum = 0.0
    for i = 0 to N - 1
        sum := sum + x[i] / N
    sum
plot(pine_sma(close, 15))

Exponentially Weighted Moving Average (EMA)

For a price series $P_t$ and look‑back length $N$ (often called the “period”):

$$\operatorname{EMA}_N(t)=\alpha P_t + (1-\alpha)\,\operatorname{EMA}_N(t-1), \qquad \alpha=\frac{2}{N+1}.$$

• Recursive form (above) is how trading platforms update the line each new bar.
• Expanded form—equivalent but shows the weights explicitly:

$$\operatorname{EMA}_N(t)=\alpha\sum_{i=0}^{\infty}(1-\alpha)^{i}P_{t-i}.$$

Because $0 < 1-\alpha < 1$, each older price is multiplied by a progressively smaller factor, so recent data dominate while the whole history still (theoretically) contributes.

indicator("ta.ema")

pine_ema(src, length) =>
    alpha = 2 / (length + 1)
    sum = 0.0
    sum := na(sum[1]) ? src : alpha * src + (1 - alpha) * nz(sum[1])
plot(pine_ema(close,15))

Form	Meaning	Typical use
na (bare constant)	“Not‑available” — the Pine equivalent of NaN/null.	Initialize a series when you deliberately want the first bar(s) to be empty.
na(x) (function)	Returns true if x is na, otherwise false.	Testing whether a previous‑bar value exists before you do math with it.

nz() means “not‑na, or zero”.

A simple moving average (SMA) assigns uniform weight inside its window and zero outside, so its lag to sudden jumps is fixed at roughly $N/2$ bars. Because EMA weights decay, its effective window shortens automatically when volatility rises: large moves add more weight to the newest bar and less to the stale tail, pulling the EMA closer to price. That “elastic” lag gives the EMA its characteristic tighter hug to the price compared with an SMA of equal period.

Why the EMA curve “looks like that”

Exponential-decay weighting

The exponential moving average (EMA) of a series $x_t$ with period $N$ can be written non-recursively as

$$\mathrm{EMA}_t=\sum_{k=0}^{\infty} w_k x_{t-k}, \quad w_k=(1-\alpha)^k \alpha, \quad \alpha=\frac{2}{N+1} .$$

The weights $w_k$ form a geometric (exponential) sequence that never quite reaches $0$, so every past sample contributes, but each step back in time is worth a constant proportion ( $1-\alpha$ ) less than the one before.

Visual effect: the curve bends smoothly toward new prices, but the bend gets shallower the farther the new price is from the old EMA, because the distant tail of small weights damps the response.

First-order low-pass filter

is the discrete-time equivalent of the differential equation of a simple RC low-pass filter:

$$\tau \frac{d y(t)}{d t}+y(t)=x(t)$$

whose impulse response is $e^{-t / \tau}$. That’s why the EMA curve has the same exponential relaxation shape when it pulls away from price spikes.

Moving Average Convergence Divergence (MACD)

The MACD is a momentum oscillator that shows the relationship between two EMAs of a security’s price.

By tracking how a short‑term EMA “converges toward” or “diverges from” a longer‑term EMA, it highlights changes in trend strength, direction, and momentum. Gerald Appel introduced it in the late 1970s, and the classic “12‑26‑9” parameters (explained below) remain the default today.

Event	Typical reading	Notes
Signal‑line crossover	DIF crosses above DEA → potential buy; crosses below → potential sell	Akin to a two‑MA crossover system but applied to DIF vs. its own average. ([Wikipedia][1])
Zero‑line crossover	DIF moves from − to + → trend turns bullish; + to − → bearish	Confirms shifts in medium‑term trend direction.
Histogram expansion / contraction	Growing bars = increasing momentum; shrinking bars = waning momentum	Gives an early visual cue before line crossovers.
Price / MACD divergence	Price makes higher highs while DIF makes lower highs (or vice‑versa)	Can foreshadow trend reversals, but often produces early signals.

https://en.wikipedia.org/wiki/MACD

$\text{MACD line} =\mathrm{EMA}_{12}-\mathrm{EMA}_{26}$

The signal line is then built as the exponential moving average of the MACD line:

$\text{Signal line} =\mathrm{EMA}_9(\text{MACD line})$

macd_val  = ta.ema(close, 12) - ta.ema(close, 26)
signal    = ta.ema(macd_val, 9)
hist      = macd_val - signal

plot(macd_val,  color=color.blue,  title="MACD")
plot(signal,    color=color.orange,title="Signal")
plot(hist,      style=plot.style_histogram, title="Histogram")