Indicators in Code: Vectorised SMA, EMA, RSI, ATR and Bollinger Bands, Verified against a Charting Platform

What you will learn

• How rolling and ewm implement SMA, EMA, Wilder smoothing, ATR and Bollinger Bands in a few lines each.
• Which parameters (adjust, alpha, ddof) silently change the numbers.
• Why your RSI or EMA differs from the charting platform in the first bars and how to compare the two correctly.
• How to organise indicators as pure functions in src/indicators.py for reuse and testing.

Explanation

The Technical Analysis in Depth course explained what these indicators mean; here the goal is correct, fast, testable code. Each function takes a Series and returns a Series on the same index.

SMA (simple moving average): def sma(s, n): return s.rolling(n).mean().

EMA (exponential moving average): def ema(s, n): return s.ewm(span=n, adjust=False).mean(). With span=n the smoothing constant is α = 2 ÷ (n + 1); for n = 20, α = 0.0952. adjust=False gives the recursive formula EMAt = α × Pt + (1 − α) × EMAt−1 that platforms use; the default adjust=True weights differently in the first few dozen bars. pandas seeds with the first price; most platforms seed with the SMA of the first n bars. Both seeds fade; after about 3n bars the series agree closely.

RSI (relative strength index), Wilder's version:

delta = s.diff()

gain = delta.clip(lower=0); loss = -delta.clip(upper=0)

avg_gain = gain.ewm(alpha=1/n, adjust=False).mean()

avg_loss = loss.ewm(alpha=1/n, adjust=False).mean()

rsi = 100 - 100 / (1 + avg_gain / avg_loss)

Wilder's smoothing is an EMA with α = 1 ÷ n, not 2 ÷ (n + 1); using span=14 here is the most common RSI bug. When avg_loss is zero, RSI correctly becomes 100.

ATR (average true range): the true range is the largest of high − low, |high − previous close| and |low − previous close|:

pc = close.shift(1)

tr = pd.concat([high - low, (high - pc).abs(), (low - pc).abs()], axis=1).max(axis=1)

atr = tr.ewm(alpha=1/n, adjust=False).mean()

Some platforms show an SMA-based ATR instead; check before comparing. Lesson 8 uses ATR for stops and sizing.

Bollinger Bands: middle = SMA(n), upper and lower = middle ± k × standard deviation over the same n bars, typically n = 20, k = 2. The trap is ddof: pandas rolling(n).std() divides by n − 1 (sample), while platforms commonly divide by n (population). Write std(ddof=0) to match. The sample value is larger, enough to turn a band "touch" into a "no touch".

Verifying against a platform. Use the same adjusted series as the platform, compute your indicator, and compare the last 100 values, because seed differences decay. Use np.allclose with a small tolerance. Persistent differences come from five usual sources: adjusted versus unadjusted data, seeding, span versus alpha, ddof, or a different input price. Fix or document the mismatch; never "tune" a parameter until the numbers agree.

Worked example

Hypothetical example: ten daily closes in euros: 20.00, 20.50, 21.00, 20.80, 21.40, 21.00, 21.60, 22.00, 21.80, 22.40.

SMA(5) on the last bar: (21.00 + 21.60 + 22.00 + 21.80 + 22.40) ÷ 5 = 108.80 ÷ 5 = €21.76.

Bollinger(5, 2) on the last bar. Deviations from 21.76: −0.76, −0.16, 0.24, 0.04, 0.64; squared sum 1.0720. Population variance = 1.0720 ÷ 5 = 0.2144, standard deviation 0.4630, upper = 21.76 + 2 × 0.4630 = €22.69, lower €20.83. Sample formula: 1.0720 ÷ 4 = 0.2680, standard deviation 0.5177, upper €22.80, lower €20.72.

EMA(5) with α = 2 ÷ 6 = 0.3333 and adjust=False: EMA1 = 20.00 (seed), EMA2 = 0.3333 × 20.50 + 0.6667 × 20.00 = 20.1667, EMA3 = 0.3333 × 21.00 + 0.6667 × 20.1667 = 20.4444, and so on to 21.8310 on the last bar.

True range for a bar with high €21.20, low €20.60 and previous close €20.80: max(0.60, |21.20 − 20.80| = 0.40, |20.60 − 20.80| = 0.20) = €0.60.

Common mistake

Practice activity

On paper, using a 3-period Wilder RSI for illustration only (real use is 14): closes 50.00, 50.50, 51.00, 50.80. (a) List the gains and losses per bar. (b) pandas ewm(alpha=1/3, adjust=False) seeds with the first observation; compute avg_gain and avg_loss on the last bar and the RSI. (c) Explain in one sentence why a platform that seeds with the simple average of the first three changes would show a different value here, and what you would do before concluding that your code is wrong.

Key takeaways

• SMA is rolling(n).mean(); EMA is ewm(span=n, adjust=False); Wilder smoothing for RSI and ATR is ewm(alpha=1/n, adjust=False).
• Bollinger Bands on most platforms use the population standard deviation: pass ddof=0.
• Seed differences fade; compare the last 100 bars with a tolerance, and look for the five usual causes of mismatch before changing code.
• Keep indicators as documented pure functions so they can be unit-tested and reused in every backtest.

Next step

Lesson 7 assembles the pieces into a vectorised backtest: signals, positions, the shift that prevents look-ahead bias, transaction costs and the equity curve.