4. Technical Indicators (for Backtesting)

Exact formulas, default periods, and the lookahead trap for each.

Algo-wide warnings.

1. Lookahead/repainting. An indicator computed on bar t's close cannot drive a trade at bar t's close in live trading. Shift the signal one bar: act at $t+1$ open, or apply signal.shift(1) to returns. Indicators that use future bars (centered MAs, ZigZag, some pivots) repaint — never backtest with them.
2. Warm-up / NaN. Every rolling indicator has an undefined warm-up window (first n bars, longer for recursive EMAs). Drop or mask it; don't trade on partially-formed values.
3. OHLC source. State whether you use close, typical price $\frac{H+L+C}{3}$, or HLC. Be consistent.
4. Adjusted prices. Use split/dividend-adjusted series, or gaps will fire false signals.

Notation: C = close, H = high, L = low, V = volume, n = period.

Trend / Moving Averages

Simple Moving Average (SMA)

Def. Unweighted mean of the last n closes. Formula. ${\text{SMA}}_t=\frac{1}{n}{\sum }_{i=0}^{n-1}{C}_{t-i}$. Defaults: 20/50/200. Signal. Price > SMA → uptrend bias. 50/200 cross = golden (bull) / death (bear) cross. Algo. C.rolling(n).mean(). NaN for first $n-1$ bars. Equal weighting lags hard on turns.

Exponential Moving Average (EMA)

Def. Weighted MA giving more weight to recent prices; reacts faster than SMA. Formula. ${\text{EMA}}_t=\alpha \cdot C_t+(1-\alpha )\cdot {\text{EMA}}_{t-1}$, with $\alpha =\frac{2}{n+1}$. Seed with SMA of first n. Signal. Same as SMA but more responsive; common 12/26 for MACD. Algo. Recursive → technically depends on all history; warm-up ~3–5×n for stability. C.ewm(span=n, adjust=False).mean().

WMA / HMA / VWMA

Def. Linearly-weighted (WMA), Hull (HMA, low-lag), volume-weighted (VWMA) variants. Formula. $\text{WMA}=\frac{\sum (w_i\cdot C_i)}{\sum w_i}$, w linear. $\text{HMA}=\text{WMA}(2\cdot \text{WMA}(n/2)-\text{WMA}(n),\sqrt{n})$. Signal. HMA reduces lag while smoothing; VWMA emphasizes high-volume prices. Algo. HMA's nested WMAs compound warm-up length; verify against a reference implementation.

MACD (Moving Average Convergence Divergence)

Def. Momentum via the gap between two EMAs. Formula. $\text{MACD}={\text{EMA}}_{12}-{\text{EMA}}_{26}$, $\text{Signal}={\text{EMA}}_9(\text{MACD})$, $\text{Histogram}=\text{MACD}-\text{Signal}$. Signal. MACD crosses above Signal → bullish; histogram = momentum strength. Divergence vs price → reversal hint. Algo. Double-recursive (EMA of EMA) — long warm-up. Crossovers lag; histogram zero-cross is earlier but noisier.

ADX / DMI (Directional Movement)

Def. Trend strength (not direction); +DI/−DI give direction. Formula. From smoothed directional movement & ATR: +DI, −DI, then $\text{DX}=\frac{100\cdot |\text{+DI}-\text{-DI}|}{\text{+DI}+\text{-DI}}$, $\text{ADX}=\text{smoothed}(\text{DX})$. Default n=14. Signal. ADX >25 → trending (use trend systems); <20 → ranging (use mean-reversion). +DI>−DI → up. Algo. Uses Wilder's smoothing (not standard EMA): $\alpha =\frac{1}{n}$. Many libs get this subtly wrong — verify.

Parabolic SAR

Def. Trailing stop-and-reverse trend follower. Formula. ${\text{SAR}}_t={\text{SAR}}_{t-1}+\text{AF}\cdot (\text{EP}-{\text{SAR}}_{t-1})$, AF (accel factor) steps 0.02→0.20 at new extremes. Signal. Dots flip side → trend reversal / stop trigger. Algo. Whipsaws badly in ranges; the AF schedule is the key parameter.

Momentum / Oscillators

RSI (Relative Strength Index)

Def. Speed/magnitude of gains vs losses, bounded 0–100. Formula. $\text{RS}=\frac{{\text{AvgGain}}_n}{{\text{AvgLoss}}_n}$ (Wilder's smoothing); $\text{RSI}=100-\frac{100}{1+\text{RS}}$. Default n=14. Signal. >70 overbought, <30 oversold (in ranges); in trends RSI can stay extreme. Divergence = reversal hint. Algo. Must use Wilder smoothing $(\alpha =1/n)$, not SMA — results differ. First value needs n+1 bars.

Stochastic Oscillator

Def. Where close sits within the recent high–low range. Formula. $\%K=\frac{100\cdot (C-L_n)}{H_n-L_n}$, $\%D={\text{SMA}}_3(\%K)$. Default n=14. Signal. >80 overbought, <20 oversold; %K crossing %D = signal. Algo. Division by zero when $H_n=L_n$ (flat range) — guard it. "Fast" vs "slow" stochastic = extra smoothing.

Rate of Change (ROC) / Momentum

Def. Percent change over n periods. Formula. $\text{ROC}=\frac{100\cdot (C_t-C_{t-n})}{C_{t-n}}$. $\text{Momentum}=C_t-C_{t-n}$. Signal. Sign = direction; magnitude = strength. Cross-sectional ROC ranks drive momentum factors. Algo. Simple, robust, no smoothing artifacts. The basis of the academic 12-1 month momentum factor.

Williams %R

Def. Inverse stochastic; position in range scaled −100..0. Formula. $\%R=\frac{-100\cdot (H_n-C)}{H_n-L_n}$. Default n=14. Signal. > −20 overbought, < −80 oversold. Algo. Same zero-range guard as stochastic.

CCI (Commodity Channel Index)

Def. Deviation of typical price from its mean, scaled. Formula. $\text{CCI}=\frac{\text{TP}-{\text{SMA}}_n(\text{TP})}{0.015\cdot \text{MeanDev}}$, $\text{TP}=\frac{H+L+C}{3}$. Default n=20. Signal. >+100 strong up / overbought; <−100 strong down / oversold. Algo. Uses mean absolute deviation (not std) — don't substitute std.

Volatility

ATR (Average True Range)

Def. Average range size including gaps; absolute volatility in price units. Formula. $\text{TR}=\max (H-L,|H-C_{prev}|,|L-C_{prev}|)$, $\text{ATR}={\text{Wilder\_avg}}_n(\text{TR})$. Default n=14. Signal. Rising ATR → expanding volatility. Used for position sizing & stops, not direction. Algo. In price units (not %), so not comparable across assets — normalize by price for cross-asset sizing. Standard for volatility-targeted stops (e.g. stop = entry − 2·ATR).

Bollinger Bands

Def. SMA envelope at ±k standard deviations. Formula. $\text{Mid}={\text{SMA}}_n(C)$, $\text{Upper/Lower}=\text{Mid}\pm k\cdot {\sigma }_n(C)$. Defaults n=20, k=2. Signal. Price at upper/lower band = stretched; band squeeze (narrow width) precedes volatility breakouts. Algo. $\%b=\frac{C-\text{Lower}}{\text{Upper}-\text{Lower}}$, $\text{bandwidth}=\frac{\text{Upper}-\text{Lower}}{\text{Mid}}$. Uses population std — match your library's ddof.

Keltner Channels

Def. EMA envelope at ±k·ATR (volatility from range, not std). Formula. $\text{Mid}={\text{EMA}}_n(C)$, $\text{Upper/Lower}=\text{Mid}\pm k\cdot \text{ATR}$. Common n=20, k=2. Signal. Bollinger inside Keltner = squeeze (low vol, breakout setup — the "TTM Squeeze"). Algo. Smoother than Bollinger (ATR < std spikes); good breakout filter.

Historical (Realized) Volatility

Def. Annualized std of returns over a window. Formula. ${\sigma }_{ann}=\text{std}(\text{log returns over }n)\times \sqrt{\text{periods per year}}$. Daily → ×√252. Signal. Regime gauge; mean-reverting. Compare to implied vol for rich/cheap options. Algo. Use log returns; pick the annualization factor matching your bar frequency (see appendix). Parkinson/Garman-Klass estimators use OHLC for efficiency.

Average Daily Range / Choppiness Index

Def. ADR = typical daily range; Choppiness = trending vs ranging gauge. Formula. $\text{Choppiness}=\frac{100\cdot {\log }_{10}\left(\frac{\sum {\text{ATR}}_n}{{\text{MaxH}}_n-{\text{MinL}}_n}\right)}{{\log }_{10}(n)}$. Signal. Choppiness >61.8 → ranging; <38.2 → trending. Pairs well with ADX as a regime filter. Algo. Bounded 0–100; useful to switch strategy mode (trend vs mean-revert).

Volume

OBV (On-Balance Volume)

Def. Cumulative volume signed by price direction. Formula. ${\text{OBV}}_t={\text{OBV}}_{t-1}+\text{sign}(C_t-C_{t-1})\cdot V_t$. Signal. OBV trend confirms/diverges from price; divergence hints at reversal. Algo. A running cumulative sum → unbounded and path-dependent; only the shape/slope matters, not the level.

VWAP (Volume-Weighted Average Price)

Def. Average price weighted by volume, typically reset each session. Formula. $\text{VWAP}=\frac{\sum ({\text{TP}}_i\cdot V_i)}{\sum (V_i)}$, $\text{TP}=\frac{H+L+C}{3}$, cumulative within the day. Signal. Institutional benchmark; price > VWAP = buyers in control intraday. Mean-reversion anchor. Algo. Resets daily — don't carry across sessions. Anchored VWAP resets from a chosen event instead.

Accumulation/Distribution & CMF

Def. Volume flow weighted by where close sits in the bar. Formula. $\text{MFV}=\frac{(C-L)-(H-C)}{H-L}\cdot V$, $\text{A/D}=\text{cumsum}(\text{MFV})$, $\text{CMF}=\frac{\sum {\text{MFV}}_n}{\sum V_n}$. Default n=20. Signal. CMF >0 → buying pressure; divergence from price = warning. Algo. Zero-range guard (H=L). CMF is bounded; A/D is cumulative (slope only).

Money Flow Index (MFI)

Def. Volume-weighted RSI. Formula. $\text{Raw MF}=\text{TP}\cdot V$, split into positive/negative by TP direction; $\text{MFI}=100-\frac{100}{1+\text{posMF}/\text{negMF}}$. Default n=14. Signal. >80 overbought, <20 oversold — like RSI but volume-aware. Algo. Needs both price and reliable volume; sparse-volume assets distort it.

Support/Resistance & Levels

Pivot Points

Def. Reference levels derived from prior period's OHLC. Formula. $P=\frac{H+L+C}{3}$, $R1=2P-L$, $S1=2P-H$, (R2/S2 extend). Computed from prior day/week. Signal. Intraday support/resistance zones. Algo. Safe (non-repainting) because they use the completed prior period — but confirm you reference the closed bar, not the forming one.

Fibonacci Retracements

Def. Pullback levels between a swing high and low. Formula. Levels at 23.6%, 38.2%, 50%, 61.8%, 78.6% of the move. Signal. Potential reversal/continuation zones. Algo. Depends on chosen swing points → subjective and can repaint if swings are auto-detected with future data. Fix the anchor or treat as discretionary.

Donchian Channels

Def. Highest high / lowest low over n periods. Formula. $\text{Upper}=\max (H_n)$, $\text{Lower}=\min (L_n)$. Default n=20/55. Signal. Breakout above upper = entry (the classic Turtle system). Algo. Use H.rolling(n).max().shift(1) so the current bar's own high doesn't define its own breakout level (subtle lookahead).

Indicator selection by regime