Building a bounded momentum oscillator with RSI smoothing

The shared bounded map

A relative strength index can be expressed as one hundred times a strength ratio divided by one plus that same ratio. In that construction, the strength ratio is the lookback average of upside closes divided by the lookback average of downside closes.

The bounded-ratio map is that transform. It converts a non-negative strength ratio into one hundred times the ratio divided by one plus the ratio.

A relative momentum index keeps the same one-hundred times ratio-over-one-plus-ratio map, but forms its ratio from smoothed multi-bar momentum increments rather than one-bar closes.

What counts as an increment

A relative strength index is a bounded oscillator formed from smoothed one-bar upside closes versus smoothed one-bar downside closes, then mapped through a ratio-to-100 transform. The upside increment is the non-negative one-bar up close. The downside increment is the non-negative absolute one-bar down close.

A relative momentum index uses the same bounded mapping and the same smoother, but each increment is the signed gap between the latest close and the close a chosen number of bars earlier. The upside momentum increment is the difference between the latest close and the close Y bars earlier when that difference is positive, and zero otherwise. The downside momentum increment is the absolute value of that same multi-bar difference when the difference is negative, and zero otherwise.

The momentum span is the lag, in bars, used to define each signed price increment before smoothing.

How the increment is smoothed

Each lookback average is updated by subtracting one lookback-fraction of the previous average and adding the latest upside close, downside close, or zero.

That step is an average-off update: a recursive smoother that removes one lookback-fraction of the previous total and adds the latest increment. The lookback is the window length that sets both the seed sum of increments and the 1-over-X decay of the recursive update.

A worked relative momentum index

A worked construction uses a five-bar momentum span and a twenty-bar smoother. After a seed window that sums the first twenty increments, later smoothed values use an average-off update: the prior total minus one-twentieth of that total plus the latest increment.

The finished oscillator value is the bounded mapping of the smoothed upside series divided by the smoothed downside series.