Tony O's Euler Bands

Tony O’s Euler Bands is a volatility-based overlay that uses the mathematical constant e (~2.71828) to scale price bands in a non-linear way. Unlike traditional Bollinger Bands or Keltner Channels, these bands are spaced by exponential functions of volatility (σ), creating zones that expand and contract more dynamically across different market regimes.

How it works:

A configurable moving average (EMA/SMA/RMA/WMA) is used as the basis.

Volatility (σ) is calculated as the standard deviation of returns over a user-defined lookback.

Four band levels are plotted above and below the basis at distances equal to:

basis × 𝑒^(𝑚⋅𝜎⋅𝑘)

where m is a user multiplier and k = {2, 4, 6, 8} for each successive band.

This produces inner bands that highlight mild deviations and outer bands that signal extreme moves.

What makes it unique:

Uses e as the base for band expansion instead of linear multiples or Fibonacci ratios.

Bands scale multiplicatively, making them more consistent across assets and price scales.

Multiple symmetric bands per side, color-coded from green (mild) to purple (extreme) for intuitive visual cues.

Optional transparent fill to show volatility envelopes without obscuring price action.

How to use:

Trend monitoring: Sustained closes beyond an inner band can indicate momentum; closes beyond outer bands can signal overextension.

Reversion spotting: Touches on extreme bands (level 4) can highlight potential exhaustion points.

Works on any asset/timeframe; adjust basis length, volatility lookback, and multiplier to suit your market.